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Linear Codes over Z_4+uZ_4: MacWilliams identities, projections, and formally self-dual codes
Abstract
Linear codes are considered over the ring Z_4+uZ_4, a non-chain extension of
Z_4. Lee weights, Gray maps for these codes are defined and MacWilliams
identities for the complete, symmetrized and Lee weight enumerators are proved.
Two projections from Z_4+uZ_4 to the rings Z_4 and F_2+uF_2 are considered and
self-dual codes over Z_4+uZ_4 are studied in connection with these projections.
Finally three constructions are given for formally self-dual codes over
Z_4+uZ_4 and their Z_4-images together with some good examples of formally
self-dual Z_4-codes obtained through these constructions.
... Furthermore, Frobenius local rings can be decomposed into their component parts and this enables us to find their generating characters. To fully understand codes over Frobenius rings, it is therefore necessary, despite the challenges, to consider local non-chain rings [5][6][7]. For more information on the subject, please refer to [5,[8][9][10][11] and the references therein. ...
... To fully understand codes over Frobenius rings, it is therefore necessary, despite the challenges, to consider local non-chain rings [5][6][7]. For more information on the subject, please refer to [5,[8][9][10][11] and the references therein. ...
... Consequently, the following theorem is established. 3,5,7,9,11,13,15, 1 + u, 3 + u, 5 + u, 7 + u, 9 + u, 11 + u, 13 + u, 15 + u}. ...
Suppose that R=Zp4[u] with u2=p3β and pu=0, where p is a prime and β is a unit in R. Then, R is a local non-chain ring of order p5 with a unique maximal ideal J=(p,u) and a residue field of order p. A linear code C of length N over R is an R-submodule of RN. The purpose of this article is to examine MacWilliams identities and generator matrices for linear codes of length N over R. We first prove that when p≠2, there are precisely two distinct rings with these properties up to isomorphism. However, for p=2, only a single such ring is found. Furthermore, we fully describe the lattice of ideals of R and their orders. We then calculate the generator matrices and MacWilliams relations for the linear codes C over R, illustrated with numerical examples. It is important to address that there are challenges associated with working with linear codes over non-chain rings, as such rings are not principal ideal rings.
... In 1963, a remarkable work of MacWilliams [25] proposed a formula that relates the weight enumerator of a code with that of its dual. Yildiz and Karadeniz [36] considered linear codes over Z 4 +uZ 4 and proved the MacWilliams identities for complete, symmetrized, and Lee weight enumerators. Aydogdu et al. [5] introduced a new class of additive codes, Z 2 Z 2 [u]-additive codes and they proved a MacWilliamstype identity on the weight enumerator of such codes. ...
... x 8 + 10x 6 + 7x 5 + 5x 3 + 12x 2 + 3, u(x 18 − 1)/(x 4 + 4x 3 + 3x + 12) [36,24,5] [ [36,12,5]] 13 17 8 ...
... x 8 + 10x 6 + 7x 5 + 5x 3 + 12x 2 + 3, u(x 18 − 1)/(x 4 + 4x 3 + 3x + 12) [36,24,5] [ [36,12,5]] 13 17 8 ...
Let be the ring of integers modulo a prime number p where is a quadratic residue modulo p. This paper presents the study of constacyclic codes over chain rings and . We also study additive constacyclic codes over and using the generator polynomials over the rings and respectively. Further, by defining Gray maps on , and we obtain some results on the Gray images of additive codes. Then we give the weight enumeration and MacWilliams identities corresponding to the additive codes over . Finally, as an application of the obtained codes, we give quantum codes using the CSS construction.
... In [20], Shi et al. have constructed two new infinite families of trace codes of dimension 2m over the ring F p + uF p , u 2 = u, where p is an odd prime. Recently, the rings such as Z 4 + uZ 4 , u 2 = 0 [24] and Z 4 + vZ 4 , v 2 = v [4,5] have been introduced to construct good codes. Yildiz and Karadeniz [24] have studied linear codes and self-dual codes over Z 4 + uZ 4 . ...
... Recently, the rings such as Z 4 + uZ 4 , u 2 = 0 [24] and Z 4 + vZ 4 , v 2 = v [4,5] have been introduced to construct good codes. Yildiz and Karadeniz [24] have studied linear codes and self-dual codes over Z 4 + uZ 4 . Cyclic codes over Z 4 + uZ 4 have been studied in [2,3,23]. ...
... Then from Theorem 2.7, we have N = 7. It can be easily verified from [24] that R has 7 distinct ideals which are: 0 , 1 , 2 , u , 2u , 2 + u , and 2, u . Example 2.9. ...
In this paper, we study cyclic codes of length n over R = Zq + uZq, u2 = 0, where q is a power of a prime p and (n; p) = 1. We have determined the complete ideal structure of R. Using this, we have obtained the structure of cyclic codes and that of their duals through the factorization of xn-1 over R. We have also computed total number of cyclic codes of length n over R. A necessary and sufficient condition for a cyclic code over R to be self-dual is presented. We have presented a formula for the total number of self-dual cyclic codes of length n over R. A new Gray map from R to Z2rp is defined. Using Magma, some good cyclic codes of length 4 over Z9 + uZ9 are obtained.
... Z 4 remain a special topic of interest in the field of algebraic coding theory because of their relation to lattices, designs, cryptography and their many applications" 1 [17]. ...
... These rings have rich mathematical theory, in particular algebraic structures. Yildiz and Karadeniz [17] derived algebraic structures related to linear codes over the ring Z 4 + uZ 4 , with u 2 = 0. They [17] also have several good formally self-dual codes over Z 4 from the codes over Z 4 + uZ 4 . ...
... Yildiz and Karadeniz [17] derived algebraic structures related to linear codes over the ring Z 4 + uZ 4 , with u 2 = 0. They [17] also have several good formally self-dual codes over Z 4 from the codes over Z 4 + uZ 4 . Bandi and Bhaintwal ( [2], [3]) considered codes over the ring Z 4 + vZ 4 , with v 2 = v, and Z 4 + wZ 4 , with w 2 = 1, 2w, respectively and derived several algebraic structures including the MacWilliams relation with respect to Rosenbloom-Tsfasman metric over the ring Z 4 + vZ 4 and the properties as well as construction of self-dual codes over the ring Z 4 + wZ 4 . ...
We investigate linear codes over the ring , with conditions , , , uv=vu, uw=wu and vw=wv. We first analyze the structure of the ring and then define linear codes over this ring. Lee weight and Gray map for these codes are defined and MacWilliams relations for complete, symmetrized, and Lee weight enumerators are obtained. The Singleton bound as well as maximum distance separable codes are also considered. Furthermore, cyclic and quasi-cyclic codes are discussed, and some examples are also provided.
... This article investigates the properties of codes over Frobenius rings. One of the main reasons that Frobenius rings are the appropriate class used to describe codes is that they satisfy MacWilliams identities, which connect the symmetrized weight enumerators of a linear code to that of its dual; see [5][6][7][8][9]. Furthermore, Frobenius local rings can be decomposed into their component parts, and this enable us to find their generating characters. ...
... This follows that | C |= 32. Take N = 4, w = (v, 0, u, v), and d = (u, v, 0, 0), which implies that | C |= 2 8 . Hence, ...
Let u,v, and w be indeterminates over Fpm and let R=Fpm+uFpm+vFpm+wFpm, where p is a prime. Then, R is a ring of order p4m, and R≅Fpm[u,v,w]I with maximal ideal J=uFpm+vFpm+wFpm of order p3m and a residue field Fpm of order pm, where I is an appropriate ideal. In this article, the goal is to improve the understanding of linear codes over local non-chain rings. In particular, we investigate the symmetrized weight enumerators and generator matrices of linear codes of length N over R. In order to accomplish that, we first list all such rings up to the isomorphism for different values of the index of nilpotency l of J, 2≤l≤4. Furthermore, we fully describe the lattice of ideals of R and their orders. Next, for linear codes C over R, we compute the generator matrices and symmetrized weight enumerators, as shown by numerical examples.
... However, to fully determine codes over Frobenius rings, it is essential to consider local rings which are not chains, even though this imposes challenges, since the rings are not PIRs. We recommend references [5][6][7][8][9][10][11][12][13][14][15] to readers. This paper mainly concentrates on determining fundamental coding results over local Frobenius rings, with a specific focus on rings of order 32, to clarify the significance of general results. ...
... From now on, suppose that R is a finite commutative singleton local ring with identity, and J denotes its Jacobson radical. We will rely on the following wellestablished results from the theory of finite rings and coding theory (see [2,3,6,8,15,19]). ...
The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In this paper, we present and successfully establish a new approach for linear codes of any finite length over local rings that are not necessarily chains. The main focus of this study is to produce generating characters, MacWilliams identities and generator matrices for codes over singleton local Frobenius rings of order 32. To do so, we first start by characterizing all singleton local rings of order 32 up to isomorphism. These rings happen to have strong connections to linear binary codes and Z4 codes, which play a significant role in coding theory.
... Extensions of the ring Z 4 received a special attention in the study of codes over rings [16,21,23,24]. For example, in a recent work ([16]) the authors consider the commutative ring Z 4 + uZ 4 + vZ 4 , where u 2 = v 2 = uv = vu = 0 with 64 elements. ...
... Whenever a new finite ring is considered in coding theory, one of the very first thing researchers do is to study cyclic codes and some of their generalizations over that ring. Among many examples of this in the literature see [11,[14][15][16][17][18][19][20][22][23][24]. One important generalization of cyclic codes is skew cyclic codes (also called θ - ...
In this paper, we study the skew-cyclic codes (also called -cyclic codes) over the ring , where . Some structural properties of the skew polynomial ring , where is an automorphism of are discussed and the elements of , the subring of fixed by , are determined. Skew cyclic codes over are viewed as left -submodules. Generator and parity-check matrices of a free -cyclic code of even length over are determined and a Gray map on is used to obtain the -images. We show that the Gray image of a free skew cyclic code over is a free linear code over . Furthermore, these codes are generalized to double skew-cyclic codes. We obtained new linear codes over from Gray images of double skew-cyclic codes over .
... After that, working on codes over rings gained greater attention. Specifically, the commutative rings of order 16 with characteristic 4 have gained greater attention [19]. One of the special properties is that, up to ring-isomorphism, the ring 44 u ZZ ( where 2 0 u and 4 u Z ) is the smallest local Frobenius commutative non-chain rings of order 16 with characteristic 4 [19]. ...
... Specifically, the commutative rings of order 16 with characteristic 4 have gained greater attention [19]. One of the special properties is that, up to ring-isomorphism, the ring 44 u ZZ ( where 2 0 u and 4 u Z ) is the smallest local Frobenius commutative non-chain rings of order 16 with characteristic 4 [19]. The interest in these rings increased after the introduction of linear and cyclic codes over the Determining the basic geometric parameter of the code, Covering Radius [5], have been an interesting task for decades. ...
This correspondence introduces the homogeneous weight and homogeneous Gray map over the ring 44 u ZZ (where 2 0 u and 4 u Z) and investigates the homogenous covering radius on 44 u ZZ of various types of repetition codes over 44 u ZZ. We also determine the covering radius of Simplex and MacDonald codes over 44 u ZZ of type and .
... The codes over finite rings remain a special topic of interest in the field of algebraic coding theory. This is because of their wide real life applications such as lattices, designs, cryptography and many others [1,2]. The study of codes over rings began in greater extent with studying codes over the ring Z 4 in [3], in particular, the work by [4,5] and subsequent literature. ...
... Let Z 4 be a commutative integer ring mod 4 with unity. Then Z 4 is a finite chain ring with the maximal ideal (2) and the characteristic 4. Let Z n 4 be the free Z 4 -module with rank n. A subset C of Z n 4 is called a linear code if and only if C is an Z 4 -submodule of Z n 4 . ...
The aim of this paper is to propose a new family of codes. We define this family over the ring , with . We derive its properties, a generator matrix and Gray images. We illustrate this new family of codes using three applications.
... In 1994, Hammons et al. [2] brought an important perspective by creating non-linear codes over F 2 from codes over Z 4 considering the Gray map. Henceforth, codes over finite rings have been widely studied [3][4][5]. ...
... Quantum error-correcting codes (QECCs) take an important role in protecting quantum information. Calderbank et al. [23] showed that QECCs can be constructed from classical error-correcting codes over GF (4). Therefore, many researchers in algebraic coding theory constructed good QECCs from classical cyclic codes over finite fields. ...
In this paper, we study cyclic codes over the ring Fp×(Fp+vFp), where p is an odd prime and v² = v. We first investigate the properties of the ring Fp×(Fp+vFp) and the linear codes over this ring. We also define a distance-preserving Gray map from Fp×(Fp+vFp) to Fp3. We discuss cyclic codes and their dual codes over the ring. Also, we define a set of generators for these codes. As an implementation, we show that quantum error-correcting codes can be obtained from dual containing cyclic codes over the ring by using the Calderbank-Shor-Steane (CSS) construction. Furthermore, we give some illustrative examples. Finally, we tabulate the non-binary quantum error-correcting codes obtained from cyclic codes over the ring.
... We can define ℛ as ℛ=ℤ 4 +uℤ 4 with u 2 =0 which is studied by Yildiz and Karadeniz in [42]. ...
... ∕ ≺u 2 ≻, which was first studied byYildiz and Karadeniz in [42]. ...
A great deal of attention has been given to codes over finite rings from the 1990s because of their new role in algebraic coding theory and their great application. One of the rings is called Frobenius ring. In this work, we characterize different properties of p-ary local Frobenius rings and their generating characters.
We find a general form for the homogeneous weights of such rings, using the generating character. In particular, we prove that the homogeneous weight has two non-zero values. We also find distance preserving, linear Gray maps for some classes of p-ary local Frobenius rings and using the Gray image, we construct many linear p-ary codes that attain the Griesmer bound
... At the same time, Kai et al. [9] have got good binary codes via Gray map. Yildiz et al. [10] have constructed the formally self-dual codes over Z 4 + uZ 4 and got the formally self-dual codes over Z 4 . Gao et al. [11] have studied the linear codes over Z 4 + vZ 4 (v 2 = v). ...
In this paper, we mainly study the theory of linear codes over the ring . By the Chinese Remainder Theorem, we have R is isomorphic to the direct sum of four rings . We define a Gray map from to , which is a distance preserving map. The Gray image of a cyclic code over is a linear code over . Furthermore, we study the MacWilliams identities of linear codes over R and give the the generator polynomials of cyclic codes over R. Finally, we discuss some properties of MDS codes over R.
... Cyclic codes over Z 4 [u]/ u k of odd length 13 As 2u k−i+t = u k−i h −1 (u i + 2u t h) ∈ C, we have u k−i+t ∈ Tor 1 (C) = u s R j , which implies k − i + t ≥ s. So we have one of the following two cases: ...
Let () where satisfies . For any odd positive integer n, it is known that cyclic codes over R of length n are identified with ideals of the ring . In this paper, an explicit representation for each cyclic code over R of length n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length n is obtained. Precisely, the dual code of each cyclic code and self-dual cyclic codes over R of length n are investigated. When k=4, some optimal quasi-cyclic codes over of length 28 and index 4 are obtained from cyclic codes over .
... Codes over local non-chain rings have received less research attention because chain rings are principal ideal rings (PIRs), and many results found for chain rings also hold for PIRs. One of the main reasons that Frobenius rings, defined later, are considered an appropriate class to characterize codes is because they meet both of the MacWilliams theorems [7][8][9]. Frobenius local rings, however, can be decomposed into their primary components. To fully comprehend codes over Frobenius rings, it is necessary, despite difficulties, to consider local non-chain rings. ...
Suppose R is a local ring with invariants p,n,r,m,k and mr=4, that is R of order p4. Then, R=R0+uR0+vR0+wR0 has maximal ideal J=uR0+vR0+wR0 of order p(m−1)r and a residue field F of order pr, where R0=GR(pn,r) is the coefficient subring of R. In this article, the goal is to improve the understanding of linear codes over small-order non-chain rings. In particular, we produce the MacWilliams formulas and generator matrices for linear codes of length N over R. In order to accomplish that, we first list all such rings up to isomorphism for different values of p,n,r,m,k. Furthermore, we fully describe the lattice of ideals in R and their orders. Next, for linear codes C over R, we compute the generator matrices and MacWilliams identities, as shown by numerical examples. Given that non-chain rings are not principal ideals rings, it is crucial to acknowledge the difficulties that arise while studying linear codes over them.
... The original results have been proven in [31] by Jessie MacWilliams and several generalizations and reformulations, through association schemes, generating functions or graphs, have appeared since [15,27,28,41,26]. Other generalizations of the MacWilliams identities have been discussed in [38] and further extensions to different code families, like convolutional codes, quantum codes and others, followed [11,20,21,9,43]. The MacWilliams identities also allow to state a Linear Programming (LP) bound, giving the tightest known upper bound on the size of a code with prescribed minimum distance and length, for an overview we refer the interested reader to [16]. ...
Famous results state that the classical MacWilliams identities fail for the Lee metric, the homogeneous metric and for the subfield metric, apart from some trivial cases. In this paper we change the classical idea of enumerating the codewords of the same weight and choose a finer way of partitioning the code that still contains all the information of the weight enumerator of the code. The considered decomposition allows for MacWilliams-type identities which hold for any additive weight over a finite chain ring. For the specific cases of the homogeneous and the subfield metric we then define a coarser partition for which the MacWilliams-type identities still hold. This result shows that one can, in fact, relate the code and the dual code in terms of their weights, even for these metrics. Finally, we derive Linear Programming bounds stemming from the MacWilliams-type identities presented.
... Since then, various weight enumerators of additive codes over Frobenius rings have been introduced, and the corresponding MacWilliams identities have been established. For instance, in [15], Hammons et al. considered ℤ 4 -additive codes, whereas Shiromoto in [26] and Tang et al. in [27] considered ℤ l -additive codes; besides, Yildiz et al. in [30] and Aydogdu et al. in [4] dealt with additive codes over ℤ 4 + uℤ 4 and ℤ 2 ℤ 2 [u] , respectively. ...
In this article, we deal with additive codes over the Frobenius ring . First, we study constacyclic codes over and and find their generator polynomials. With the help of these generator polynomials, we determine the structure of constacyclic codes over . We use Gray maps to show that constacyclic codes over are essentially binary generalized quasi-cyclic codes. Moreover, we obtain a number of binary codes with good parameters from these -constacyclic codes. Besides, several weight enumerators are computed, and the corresponding MacWilliams identities are established.
... Te structure of a cyclic code of arbitrary length over the ring Z p [u, v]/〈u 2 , v 2 , uv − vu〉 has been determined by Parmod Kumar Kewat et al. [16]. Linear and cyclic codes over the nonchain ring Z 4 + uZ 4 , u 2 � 0 were frst introduced by Yildiz et al. [17,18]. Tey have found some good linear codes over Z 4 as the Gray images of cyclic codes over Z 4 + uZ 4 , u 2 � 0. Te structure of a cyclic code of arbitrary length over Z 4 + uZ 4 , u 2 � 0 has been studied by Bandi and Bhaintwal [19]. ...
The rings Z 4 + ν Z 4 have been classified into chain rings and nonchain rings based on the values of ν 2 ∈ Z 4 + ν Z 4 . In this paper, the structure of a cyclic code of arbitrary length over the rings Z 4 + ν Z 4 for those values of ν 2 for which these are nonchain rings has been established. A unique form of generators for a cyclic code over these rings has also been obtained. Furthermore, the rank and cardinality of a cyclic code over these rings have been established by finding a minimal spanning set for the code.
... Due to the wide-ranging applications of codes over Z 4 ([27], [17]), such as in cryptography ( [28], [15]), DNA coding ( [21], [12]), and quaternary modulation ( [20], [23]), there has been significant research devoted to studying linear and cyclic codes over its various extension rings, including Z 4 +uZ 4 ( [4], [31], [30], [6], [19], [11], [10]). Due to the importance of codes over Z 4 , a database of Z 4 codes was created. ...
... One of the extensions of Z 4 of order 16 is the ring Z 4 + uZ 4 with u 2 = 1. Codes over this ring have been studied recently [20,23,26,2]. This ring can be written as Z 4 [u]/⟨u 2 − 1⟩ and it is isomorphic to Z 4 [u]/⟨u 2 + 2u⟩ by the map u → u + 1. ...
... Linear and cyclic codes over the non-chain ring Z 4 + νZ 4 , ν 2 = 0 were first introduced by Yildiz et al. [22,19]. The structure of cyclic codes of arbitrary length over Z 4 + νZ 4 , ν 2 = 0 has been studied by Bandi and Bhaintwal [6]. ...
The rings Z 4 + νZ 4 have been classified into chain rings and non-chain rings on the basis of the values of ν 2 ∈ Z 4 + νZ 4. In this paper, the structure of cyclic codes of arbitrary length over the rings Z 4 + νZ 4 for those values of ν 2 for which these are non-chain rings has been established. A unique form of generators of these codes has also been obtained. Further, rank and cardinality of these codes have been established by finding minimal spanning sets for these codes.
... Various weight enumerators were investigated over Z l in [22,23] and certain conditions that ensures the existence of such an identity were discussed. In 2014, the authors investigated in [25] linear codes over Z 4 + uZ 4 and their weight enumerators. In [1], the authors introduced Z 2 Z 2 [u]-additive codes and obtained the generator matrices in the standard form. ...
Additive complementary dual (in short, ACD) codes are considered over the ring Z2R=Z2×Z2[u]⟨u4⟩. We investigate free self-dual codes over R. A condition that ensures an additive code to be an ACD code is established. Furthermore, for a separable additive code to be an ACD code, a necessary and sufficient condition is obtained. We study a Gray map under which certain additive codes become binary linear complementary dual (in short, LCD) codes. We also present a few optimal (or almost optimal) binary LCD codes. Moreover, a number of weight enumerators are computed and the corresponding MacWilliams identities are discussed.
... Motivated by the approach, in 2014 linear codes as well as self dual codes which are analogous allowing some projections from the ring Z 4 + uZ 4 to Z 4 and F 2 + uF 2 , have been constructed by Yildiz et. al [14]. Furthermore, MacWilliams identities were also established in this paper. ...
In this present work, we generalize the study of construction of DNA codes over the rings , for . Rigorous study along with characterization of the ring structures is presented. We extend the Gau map and Gau distance, defined in \cite{DKBG}, over all the 16 rings . Furthermore, an isometry between the codes over the rings and the analogous DNA codes is established in general. Brief study of dual and self dual codes over the rings is given including the construction of special class of self dual codes that satisfy reverse and reverse-complement constraints. The technical contributions of this paper are twofold. Considering the Generalized Gau distance, Sphere Packing-like bound, GV-like bound, Singleton like bound and Plotkin-like bound are established over the rings . In addition to this, optimal class of codes are provided with respect to Singleton-like bound and Plotkin-like bound. Moreover, the construction of family of DNA codes is proposed that satisfies reverse and reverse-complement constraints using the Reed-Muller type codes over the rings .
... The MacWilliams identity has explained to the residue class of integers module k and with respect to the Lee weight of linear codes over Z 4 was determined; see [8]. Also, the MacWilliams identity has been studied for linear codes over Z 4 + uZ 4 and F 2 + uF 2 + vF 2 with respect to the Lee weight; (see [3,10]). This paper is organized as follows: In Section 2, a study of linear codes over a chain ring and a method for finding the MacWilliams identity with homogeneous weight are presented. ...
... After that, working on the codes over rings gained greater attention. What started with the ring Z 4 , was later generalized to the rings Z 2 s , Z 2 + uZ 2 , Z 4 + uZ 4 , F p + uF p etc. [5][6][7][8]. Covering Radius is a widely discussed parameter for the codes with respect to the Hamming weight [9]. A lot of other weights such as Lee weight [4], Homogenous weight [10] and Euclidean weight have been introduced and used in the literature for convenience. ...
In this correspondence, we investigate the covering radius of various types of repetition codes over Zpk(k≥2) with respect to the Lee distance. We determine the exact covering radius of the various repetition codes, which have been constructed using the zero divisors and units in Zpk. We also derive the lower and upper bounds on the covering radius of block repetition codes over Zpk.
... Then Dougherty et al. [20] investigated self-dual codes over commutative Frobenius rings. In [28] and [29], Yildiz et al. studied codes over an extension ring of Z 4 and obtained some good Z 4 -codes. Here the ring was described as Z 4 [u]/ u 2 = Z 4 +uZ 4 (u 2 = 0) which is a local non-principal ring. ...
In this paper, an explicit representation and enumeration for negacyclic codes of length 2^kn over the local non-principal ideal ring R=Z_4+uZ_4 (u^2=0) is provided, where k, n are any positive integers and n is odd. In particular, all distinct negacyclic codes of length 2^k over R are listed precisely. Moreover, an exact mass formula for the number of negacyclic codes of length 2^kn over R is given and some mistakes in [Cryptogr. Commun. (2017) 9: 241--272] are corrected.
... A non-chain extension of Z 4 was taken by Yildiz and Karadeniz in [26]. They discussed the linear codes over Z 4 + uZ 4 , where u is an indeterminate which satisfies u 2 = 0. Formally self-dual codes and their applications to real and complex lattices were also studied by them. ...
Let R=GR(pe,m)[u]∕〈uk〉 be a finite commutative ring for a prime p and any positive integers e,m and k. In this paper, we derive the explicit representation of cyclic codes over the ring R of length n, where n and p are coprime. We also discuss the dual of such cyclic codes over the ring R and give a sufficient condition for the codes to be self-dual. Moreover, we study quasi-cyclic codes of length kn and index k over the ring R, and obtain some good codes satisfying the bound given in Dougherty and Shiromoto (2000) over the ring Z9 as an example.
... The certain type of codes over many …nite rings were studied [2,4,5,8,9,13,15,16,20,21,22]. Many of good codes were obtained from them. ...
... In this section, we produce a generator matrix for a code over any local Frobenius non-chain ring of order 16, these being the smallest local Frobenius non-chain rings. The local Frobenius non-chain rings of order 16 that have been studied are F 2 [u, v]/ u 2 , v 2 (see [1,2,10]), Z 4 [x]/ x 2 (see [11]) and Z 4 [x]/ x 2 − 2x (see [6]). None of these works have given a standard generator matrix. ...
Codes over commutative Frobenius rings are studied with a focus on local Frobenius rings of order 16 for illustration. The main purpose of this work is to present a method for constructing a generating character for any commutative Frobenius ring. Given such a character, the MacWilliams identities for the complete and symmetrized weight enumerators can be easily found. As examples, generating characters for all commutative local Frobenius rings of order 16 are given. In addition, a canonical generator matrix for codes over local non-chain rings is discussed. The purpose is to show that when working over local non-chain rings, a canonical generator matrix exists but is less than useful which emphases the difficulties in working over such rings.
... In the case of Z 4 +uZ 4 , the Lee weight was defined in [12] as shown in Table 1. . We will adopt a similar technique here. ...
... Then self-dual codes over commutative Frobenius rings were investigated by Dougherty et al. [16]. Especially, in 2014, codes over an extension ring of Z 4 were studied by Yildiz et al. [25] and [26], here the ring was described as Z 4 [u]/ u 2 = Z 4 + uZ 4 (u 2 = 0) which is a local non-principal ring. Then a complete classification and an explicit representation for cyclic codes of odd length over Z 4 [u]/ u k = Z 4 + uZ 4 + . . . ...
In this paper, an explicit representation and enumeration for negacyclic codes of length over the local non-principal ideal ring is provided, where are any positive integers and n is odd. As a corollary, all distinct negacyclic codes of length over R are listed precisely. Moreover, a mass formula for the number of negacyclic codes of length over R is given and a mistake in [Cryptogr. Commun. (2017) 9: 241--272] is corrected.
... Then self-dual codes over commutative Frobenius rings were investigated by Dougherty et al. [16]. Especially, in 2014, codes over an extension ring of Z 4 were studied by Yildiz et al. [25] and [26], here the ring was described as Z 4 [u]/ u 2 = Z 4 + uZ 4 (u 2 = 0) which is a local non-principal ring. Then a complete classification and an explicit representation for cyclic codes of odd length over Z 4 [u]/ u k = Z 4 + uZ 4 + . . . ...
In this paper, an explicit representation and enumeration for nega-cyclic codes of length 2 k n over the local non-principal ideal ring R = Z 4 + uZ 4 (u 2 = 0) is provided, where k, n are any positive integers and n is odd. As a corollary, all distinct negacyclic codes of length 2 k over R are listed precisely. Moreover, a mass formula for the number of negacyclic codes of length 2 k n over R is given and a mistake in [Cryptogr. Commun. (2017) 9: 241-272] is corrected.
... Codes over rings have generated a lot of interest after a breakthrough paper by Hammons et al. [15] in 1994. Recently, many extension rings of Z 4 have been considered by researchers to construct codes [24,25]. In most of these studies, cyclic codes have been studied in commutative settings. ...
In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over R = ℤ 4 + uℤ 4 ; u ² = 1, with an automorphism θ and a derivation δ θ . We generalize the notion of cyclic codes to skew-cyclic codes with derivation, and call such codes as δ θ -cyclic codes. Some properties of skew polynomial ring R[x, θ, δ θ ] are presented. A δ θ -cyclic code is proved to be a left R[x, θ, δ θ ]-submodule of R[x,θ,δ〈xn−1〉θ] . The form of a parity-check matrix of a free δ θ -cyclic codes of even length n is presented. These codes are further generalized to double δ θ -cyclic codes over R. We have obtained some new good codes over ℤ 4 via Gray images and residue codes of these codes. The new codes obtained have been reported and added to the database of ℤ 4 -codes [2].
... Then self-dual codes over commutative Frobenius rings were investigated by Dougherty et al. [13]. Especially, in 2014, codes over an extension ring of Z 4 were studied in [24] and [25], here the ring was described as Z 4 [u]/ u 2 = Z 4 + uZ 4 (u 2 = 0) which is a local non-principal ring. ...
Let () and n be an odd positive integer.
Then R is a local non-principal ideal ring of 16 elements and there is a -linear Gray map from R onto which preserves Lee distance and orthogonality.
First, a canonical form decomposition and the structure for any negacyclic code over R of length 2n are presented. From this decomposition, a complete classification of all these codes is obtained. Then the cardinality and the dual code
for each of these codes are given, and self-dual negacyclic codes over R of length 2n are presented. Moreover, all
negacyclic codes over R of length and all self-dual codes among them are presented precisely, where is a Mersenne prime. Finally, 36 new and good self-dual 2-quasi-twisted linear codes over with basic parameters and of type and basic parameters and of type which are Gray images of self-dual negacyclic codes over R of length 14 are listed.
... Odd length DNA codes was given which satisfies the Hamming distance constraints from the commutative ring F 2 [u]/ < u 4 − 1 > with u 4 = 1 in [12]. A new ring Z 4 + uZ 4 with 16 elements was introduced by Yildiz et al. in [13] and DNA codes of odd lengths from the ring were studied in [14]. ...
In this work, we study the DNA codes from the ring R = Z4 + wZ4, where w^2 = 2+2w with 16 elements. We establish a one to one correspondence between the elements of the ring R and all the DNA codewords of length 2 by defining a distance preserving Gau map phi. Using this map, we give several new classes of the DNA codes which satisfies reverse and reverse complement constraints. Some of the constructed DNA codes are optimal.
... The generating character of R is given by χ(a + ub) = e πi 2 (a+b) = i a+b. It can be shown that R is isomorphic to the quotient ring Z 4 [x]/x 2. Yildiz and Karadeniz in [6] emulated the Gray map Z 4 → F 2 2 to define the map φ 3 : R → Z 2 4 by φ 3 (a + ub) = (b, a + b), and then extended the classical Lee weight on Z 4 to R by w L (a + ub) = w L (b, a + b) = w L (b) + w L (a + b). However, this weight is not homogeneous. ...
Let F2 be the binary field and Z_{2^r} the residue class ring of integers modulo 2^r , where r is a positive integer. For the finite 16-element commutative local Frobenius non-chain ring Z4+uZ4, where u is nilpotent of index 2, two weight functions are considered, namely the Lee weight and the homogeneous weight. With the appropriate application of these weights, isometric maps from Z4+uZ4 to the binary spaces F2^4 and F2^8 , respectively, are established via the composition of other weight-based isometries. The classical Hamming weight is used on the binary space. The resulting isometries are then applied to linear block codes over Z4+uZ4 whose images are binary codes of predicted length, which may or may not be linear. Certain lower and upper bounds on the minimum distances of the binary images are also derived in terms of the parameters of the Z4+uZ4 codes. Several new codes and their images are constructed as illustrative examples. An analogous procedure is performed successfully on the ring Z8+uZ8, where u ² = 0, which is a commutative local Frobenius non-chain ring of order 64. It turns out that the method is possible in general for the class of rings Z_{2^r}+uZ_{2^r} , where u ² = 0, for any positive integer r, using the generalized Gray map from Z_{2^r} to (F_2}^{2^{r-1}}.
... Then self-dual codes over commutative Frobenius rings were investigated in Dougherty et al. [11]. Especially, in 2014, codes over an extension of Z 4 were studied in Yildiz et al. [19] and [20] where many good Z 4 -codes were obtained as images. The ring in the mentioned works was described as Z 4 [u]/⟨u 2 ⟩ = Z 4 + uZ 4 (u 2 = 0) which is a local non-principal ring. ...
Let p be a prime integer, n, s ≥ 2 be integers satisfying gcd(p, n) = 1, and denote R = Z p s [v]/⟨v 2 − pv⟩. Then R is a local non-principal ideal ring of p 2s elements. First, the structure of any cyclic code over R of length n and a complete classification of all these codes are presented. Then the cardinality of each code and dual codes of these codes are given. Moreover, self-dual cyclic codes over R of length n are investigated. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes over Z 4 of length 30 and extremal 4-quasi-cyclic self-dual binary linear [60, 30, 12] codes derived from cyclic codes over Z 4 [v]/⟨v 2 + 2v⟩ of length 15.
... Then self-dual codes over commutative Frobenius rings were investigated in Dougherty et al. [13]. Especially, codes over an extension rings of Z 4 were studied in Yildiz et al. [25] and [26] where many good Z 4 -codes were obtained as images. The ring in the mentioned works was described as Z 4 [u]/ u 2 = Z 4 + uZ 4 (u 2 = 0) which is a local non-principal ring. ...
Let p be a prime integer, be integers satisfying , and denote . Then R is a local non-principal ideal ring of elements. First, the structure of any cyclic code over R of length n and a complete classification of all these codes are presented. Then the cardinality of each code and dual codes of these codes are given. Moreover, self-dual cyclic codes over R of length n are investigated. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes over of length 30 and extremal 4-quasi-cyclic self-dual binary linear [60,30,12] codes derived from cyclic codes over of length 15.
The purpose of this manuscript is two-fold. First, properties of the ring and the set of ideals are established. Second, results on cyclic codes of length n, , over the non-chain Frobenius ring and their description by means of idempotent elements are presented.
In this paper, we investigate the structure and properties of skew negacyclic codes and skew quasi-negacyclic codes over the ring [Formula: see text] Some structural properties of [Formula: see text] are discussed, where [Formula: see text] is an automorphism of [Formula: see text] A skew quasi-negacyclic code of length [Formula: see text] with index [Formula: see text] over [Formula: see text] is viewed both as in the conventional row circulant form and also as an [Formula: see text]-submodule of [Formula: see text], where [Formula: see text] is the Galois extension ring of degree [Formula: see text] over [Formula: see text] and [Formula: see text] is an automorphism of [Formula: see text] A sufficient condition for one generator skew quasi-negacyclic codes to be free is determined. Some distance bounds for free one generator skew quasi-negacyclic codes are discussed. Furthermore, given the decomposition of a skew quasi-negacyclic code, we provide the decomposition of its dual code. As a result, a characterization of xself-dual skew quasi-negacyclic codes over [Formula: see text] is provided. By using computer search we obtained a number of new linear codes over [Formula: see text] from skew negacyclic and skew quasi-negacyclic codes over [Formula: see text].
Let R=Z4[v]/〈v2+2v〉=Z4+vZ4 (v2=2v) and n be an odd positive integer. Then R is a local non-principal ideal ring of 16 elements and there is a Z4-linear Gray map from R onto Z42 which preserves Lee distance and orthogonality. First, a canonical form decomposition and the structure for any negacyclic code over R of length 2n are presented. From this decomposition, a complete classification of all these codes is obtained. Then the cardinality and the dual code for each of these codes are given, and self-dual negacyclic codes over R of length 2n are presented. Moreover, all 23⋅(4p+5⋅2p+9)2p−2p negacyclic codes over R of length 2Mp and all 3⋅(4p+5⋅2p+9)2p−1−1p self-dual codes among them are presented precisely, where Mp=2p−1 is a Mersenne prime. Finally, 36 new and good self-dual 2-quasi-twisted linear codes over Z4 with basic parameters (28,228,dL=8,dE=12) and of type 21447 and basic parameters (28,228,dL=6,dE=12) and of type 21646 which are Gray images of self-dual negacyclic codes over R of length 14 are listed.
In this paper, we develop a theory for constructing cyclic codes of odd length over the ring [Formula: see text], where [Formula: see text], which plays an important role in DNA computing. A direct link between the elements of [Formula: see text] and the [Formula: see text] codons used in the amino acids of the living organisms is established. Then, we investigate reversible cyclic codes and reversible complement cyclic codes of odd length over [Formula: see text]. Moreover, we give some properties of binary images of the codons under the Gray map. Finally, two examples of cyclic codes over [Formula: see text] with their minimum Hamming distance will be studied as well.
Linear codes and a Gray map are considered over the non-chain Frobenius ring R_m = Z_{2^{m}} + u Z_{2^{m}}. The complete Lee weight enumerator for these codes is defined and the MacWilliams Identity is presented.
In this paper, we investigate Smarandache curves according to type-2 Bishopframe in Euclidean 3- space and we give some differential geometric properties of Smarandache curves. Also, some characterizations of Smarandache breadth curves in Euclidean 3-space are presented. Besides, we illustrate examples of our results.
We study α-constacyclic codes over the Frobenius non-chain ring R≔Z4[u]∕〈u2−1〉 for any unit α of R. We obtain new MDR cyclic codes over Z4 using a close connection between α-constacyclic codes over R and cyclic codes over Z4. We first explicitly determine generators of all α-constacyclic codes over R of odd length n for any unit α of R. We then explicitly obtain generators of cyclic codes over Z4 of length 2n by using a Gray map associated with the unit α. This leads to a construction of infinite families of MDR cyclic codes over Z4, where a MDR code means a maximum distance with respect to rank code in terms of the Hamming weight or the Lee weight. We obtain 202 new cyclic codes over Z4 of lengths 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50 and 54 by implementing our results in Magma software; some of them are also MDR codes with respect to the Hamming weight or the Lee weight.
In this work, we study linear codes over the ring and their weight enumerators, where . We first give the structure of the ring and investigate linear codes over this ring. We also define two weights called Lee weight and Gray weight for these codes. Then we introduce two Gray maps from to and study the Gray images of linear codes over the ring. Moreover, we prove MacWilliams identities for the complete, the symmetrized and the Lee weight enumerators.
In the present paper, we study the structure of cyclic DNA codes of even length over the ring where . We investigate two presentations of cyclic codes of even length over satisfying the reverse constraint and the reverse-complement constraint.
In this paper, we study λ-constacyclic codes over the ring R = Z4 + uZ4, where u² = 0, for λ =1 + 3u and 3 + u. We introduce two new Gray maps from R to Z4⁴ and show that the Gray images of λ-constacyclic codes over R are quasi-cyclic over Z4. Moreover, we present many examples of λ-constacyclic codes over R whose Z4-images have better parameters than the currently best-known linear codes over Z4. © 2018 Academy of Mathematics and Systems Science, Chinese Academy of Sciences.
We develop the theory for constructing DNA cyclic codes of odd length over R=Z4[u]/〈u²−1〉 based on the deletion distance. Cyclic codes of odd length over R satisfying the reverse constraint and the reverse-complement constraint are discussed. The GC-content of these codes and their deletion distance are studied. Among others, examples of cyclic DNA codes with GC-content and their respective deletion distance are provided.
For over a decade, there has been considerable research on codes over ℤ 4 and other rings. In spite of this, no tables or databases exist for codes over ℤ 4 , as it is the case with codes over finite fields. The purpose of this work is to contribute to the creation of such a database. We consider cyclic, negacyclic and quasi-twisted (QT) codes over ℤ 4 . Some of these codes have binary images with better parameters than the best-known binary linear codes. We call such codes “good codes”. Among them are two codes which improve the bounds on the best-known binary nonlinear codes. Tables of best cyclic and QT codes over ℤ 4 are presented.
Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z_4 domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z_4 -- and the Nordstrom-Robinson code is self-dual -- which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z_4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z_4, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z_4, but extended Hamming codes of length n >= 32 and the Golay code are not. Using Z_4-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code.
The conditions satisfied by the weight enumerator of self-dual codes, defined over the ring of integers module four, have been studied by Klemm (1989), then by Conway and Sloane (1993). The MacWilliams (1977) transform determines a group of substitutions, each of which fixes the weight enumerator of a self-dual code. This weight enumerator belongs to the ring of polynomials fixed by the group of substitutions, called the ring R of invariants. Among all of the quaternary self-dual codes, some have the property that all euclidean weights are multiples of 8. These codes are called type II codes by analogy with the binary case. An upper bound on their minimum euclidean weight is given, thereby leading to a natural notion of extremality akin to similar concepts for type II binary codes and type II lattices. The most interesting examples of type II codes are perhaps the extended quaternary quadratic residue codes. This class of codes includes the octacode [8, 4, 6] and the lifted Golay [24, 12, 12]. Other classes of interest comprise a multilevel construction from binary Reed-Muller and lifted double circulant codes
The alphabet F2+uF2 is viewed here as a
quotient of the Gaussian integers by the ideal (2). Self-dual F<sub>2
</sub>+uF2 codes with Lee weights a multiple of 4 are called
Type II. They give even unimodular Gaussian lattices by Construction A,
while Type I codes yield unimodular Gaussian lattices. Construction B
makes it possible to realize the Leech lattice as a Gaussian lattice.
There is a Gray map which maps Type II codes into Type II binary codes
with a fixed point free involution in their automorphism group.
Combinatorial constructions use weighing matrices and strongly regular
graphs. Gleason-type theorems for the symmetrized weight enumerators of
Type II codes are derived. All self-dual codes are classified for length
up to 8. The shadow of the Type I codes yields bounds on the highest
minimum Hamming and Lee weights
Type II Z 4-codes are introduced as self-dual
codes over the integers modulo 4 containing the all-one vector and with
Euclidean weights multiple of 8. Their weight enumerators are
characterized by means of invariant theory. A notion of extremality for
the Euclidean weight is introduced. Their binary images under the Gray
map are formally self-dual with even weights. Extended quadratic residue
Z 4-codes are the main example of this family of
codes. They are obtained by Hensel lifting of the classical binary
quadratic residue codes. Their binary images have good parameters. With
every type II Z 4-code is associated via construction
A modulo 4 an even unimodular lattice (type II lattice). In dimension
32, we construct two unimodular lattices of norm 4 with an automorphism
of order 31. One of them is the Barnes-Wall lattice BW32
Certain notorious nonlinear binary codes contain more codewords
than any known linear code. These include the codes constructed by
Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals
(1974), and Delsarte-Goethals (1975). It is shown here that all these
codes can be very simply constructed as binary images under the Gray map
of linear codes over Z 4, the integers mod 4
(although this requires a slight modification of the Preparata and
Goethals codes). The construction implies that all these binary codes
are distance invariant. Duality in the Z 4 domain
implies that the binary images have dual weight distributions. The
Kerdock and “Preparata” codes are duals over
Z 4-and the Nordstrom-Robinson code is
self-dual-which explains why their weight distributions are dual to each
other. The Kerdock and “Preparata” codes are
Z 4-analogues of first-order Reed-Muller and extended
Hamming codes, respectively. All these codes are extended cyclic codes
over Z 4, which greatly simplifies encoding and
decoding. An algebraic hard-decision decoding algorithm is given for the
“Preparata” code and a Hadamard-transform soft-decision
decoding algorithm for the I(Kerdock code. Binary first- and
second-order Reed-Muller codes are also linear over Z <sub>4
</sub>, but extended Hamming codes of length n⩾32 and the Golay code
are not. Using Z 4-linearity, a new family of
distance regular graphs are constructed on the cosets of the
“Preparata” code
New 5-designs on 24 points were constructed recently by Harada by the consideration of Z4-codes. We use Jacobi polynomials as a theoretical tool to explain their existence as resulting of properties of the symmetrized weight enumerator (swe) of the code. We introduce the notion of a colored t-design and we show that the words of any given Lee composition, in any of the 13 Lee-optimal self-dual codes of length 24 over Z4, form a colored 5-design. New colored 3-designs on 16 points are also constructed in that way.
. This paper sets a foundation for the study of linear codes over finite rings. The finite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from finite fields to finite Frobenius rings. It is over Frobenius rings that certain key identifications can be made between the ring and its complex characters. Introduction Since the appearance of [8] and [19], using linear codes over Z=4Z to explain the duality between the non-linear binary Kerdock and Preparata codes, there has been a revival of interest in codes defined over finite rings. This paper examines the foundations of algebraic coding theory over finite rings and singles out the finite Frobenius rings as the most appropriate rings for coding theory. Why are Frobenius rings appropriate for coding theory? Because two classical theorems of MacWilliams---the extension theorem and th...
We give families of self-dual ℤ 4 -codes of Type IV-I and Type IV-II generated by conference matrices and skew-Hadamard matrices. Furthermore, we give a family of self-dual ℤ 4 -codes of Type IV-I generated by bordered skew-Hadamard matrices.
New 5-designs on 24 points were constructed recently by Harada by the consideration of ℤ 4 -codes. We use Jacobi polynomials as a theoretical tool to explain their existence as resulting of properties of the symmetrized weight enumerator of the code. We introduce the notion of a colored t-design and we show that the words of any given Lee composition, in any of the 13 Lee-optimal self-dual codes of length 24 over ℤ 4 , form a colored 5-design. New colored 3-designs on 16 points are also constructed in that way.
Fornodd, theZ"4cyclic code generated by 2 is self-dual. We call this a trivial cyclic self-dual code. When do there exist nontrivial cyclic self-dual codes of odd lengthn? We give an answer in this paper by characterizing thesenand describing generators of such codes; this yields an existence test for cyclic difference sets. We also give all examples of nontrivial cyclic self-dual codes up to length 39. From these nontrivial cyclic, self-dual codes, construction A yields unimodular lattices of Type I, some of which are extremal; extension and augmentation yields three new extremal Type II codes of length 32, and an extremal self-dual code of Type II of length 40.
We introduce codes over an infinite family of rings and describe two Gray maps to binary codes which are shown to be equivalent. The Lee weights for the elements of these rings are described and related to the Hamming weights of their binary image. We describe automorphisms in the binary image corresponding to multiplication by units in the ring and describe the ideals in the ring, using them to define a type for linear codes. Finally, Reed Muller codes are shown as the image of linear codes over these rings.
In this work, we investigate linear codes over the ring
. We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43–65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32–45, 1999). We then characterize the
-linearity of binary codes under the Gray map and give a main class of binary codes as an example of
-linear codes. The duals and the complete weight enumerators for
-linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over
are obtained.
According to V. Pless et al. (1997, J. Combin. Theory Ser. A78, 32–50) all Z4 codes of type II and length 16 are known. In this note we relate the five optimal codes to the octacode. We also construct an optimal quaternary iso-dual [14, 7, 8] code which was not known previously.
Fundamentals of Error Correcting Codes is an in-depth introduction to coding theory from both an engineering and mathematical viewpoint. As well as covering classical topics, there is much coverage of techniques which could only be found in specialist journals and book publications. Numerous exercises and examples and an accessible writing style make this a lucid and effective introduction to coding theory for advanced undergraduate and graduate students, researchers and engineers, whether approaching the subject from a mathematical, engineering or computer science background.
A lift of binary self-dual codes to the ring R2 is described. By
using this lift, a family of self-dual codes over R2 of length 17 are constructed.
Taking the binary images of these codes, extremal binary self-dual codes of
length 68 are obtained. For the �rst time in the literature, extremal binary
codes of length 68 with
= 4 and
= 6 in W68;2 have been obtained. In
addition to these, six new codes with
= 0 and fourteen new codes with
= 2
in W68;2 have also been found.
In this work, the double-circulant, bordered-double-circulant and stripped bordered-double-circulant constructions for self-dual codes over the non-chain ring R 2 = F 2 + uF 2 + vF 2 + uvF 2 are introduced. Using these methods, we have constructed three extremal binary Type I codes of length 64 of new weight enumerators for which extremal codes were not known to exist. We also give a double-circulant construction for extremal binary self-dual codes of length 40 with covering radius 7.
We introduce codes over the ring
We relate self-dual codes over this ring to quaternionic unimodular lattices and to self-dual codes over
via a gray map. We study a connection between the complete weight enumerators of codes over the quaternionic ring Σ2m
and Jacobi forms over the half-space of quaternions. This motivates us to construct an algebra homomorphism from a certain invariant polynomial ring, where the complete weight enumerators belong, to the ring of Jacobi forms over the quaternions. Higher genus modular forms over the quaternions are also constructed using joint weight enumerators of codes.
Recently,an optimal formally self-dual Z4-code of length 14 and minimum Lee weight 6 has been found using the double circulant
construction by Duursma,Greferath and Schmidt. In this paper,we classify all optimal double circulant Z4-codes up to length
32. In addition, double circulant codes with the largest minimum Lee weights for this class of codes are presented for lengths
up to 32.
We introduce the finite ring . We develop a theory of self-dual codes over this ring and relate self-dual codes over this ring to complex unimodular lattices. We describe a theory of shadows for these codes and lattices. We construct a gray map from this ring to the ring and relate codes over these rings, giving special attention to the case when m=2. We construct various Hermitian modular forms from weight enumerators and give the correspondence between the invariant space, where the weight enumerators of codes reside, and the space of Hermitian modular forms.
The Assmus-Mattson theorem is a method to find designs in linear
codes over a finite field. The purpose of this paper is to give an
analog of this theorem for Z4-codes by using the harmonic
weight enumerator introduced by Bachoc. This theorem can find some
5-designs in the lifted Golay code over Z4 which were
discovered previously by other methods
The negashift ν of Z4n is defined as the
permutation of Z4n such that ν(a0, a
1, ···, ai,
···, an-1)=(-an-1, a<sub>0
</sub>, ···, ai,
···, an-2) and a negacyclic code of
length n over Z4 is defined as a subset C of Z4
n such that ν(C)=C. We prove that the Gray image of a
linear negacyclic code over Z4 of length n is a binary
distance invariant (not necessary linear) cyclic code. We also prove
that, if n is odd, then every binary code which is the Gray image of a
linear cyclic code over Z4 of length n is equivalent to a
(not necessary linear) cyclic code and this equivalence is explicitely
described. This last result explains and generalizes the existence,
already known, of versions of Kerdock, Preparata, and others codes as
doubly extended cyclic codes. Furthermore, we introduce a family of
binary linear cyclic codes which are Gray images of Z4 linear
negacyclic codes
In previous work by Huffman and by Yorgov (1983), a decomposition
theory of self-dual linear codes C over a finite field Fq was
given when C has a permutation automorphism of prime order r relatively
prime to q. We extend these results to linear codes over the Galois ring
Z 4 and apply the theory to
Z 4-codes of length 24. In particular we obtain 42
inequivalent [24,12] Z 4-codes of minimum Euclidean
weight 16 which lead to 42 constructions of the Leech lattice
- Z X Wan
Z.X. Wan, Series on Applied Mathematics: Quaternary Codes, World Scientific, 1997.
Codes over R k , Gray maps and their binary images, Finite Fields Appl
- S T Dougherty
- B Yildiz
- S Karadeniz
S.T. Dougherty, B. Yildiz, S. Karadeniz, Codes over R k, Gray maps and their binary images, Finite
Fields Appl. 17 (2011) 205–219.