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Constructing formally self-dual codes over RkRk
Abstract
In this work, we study construction techniques of formally self-dual codes over the infinite family of rings Rk=F2[u1,u2,…,uk]/〈ui2=0,uiuj=ujui〉. These codes give rise to binary formally self-dual codes. Using these constructions, we obtain a number of good formally self-dual binary codes including even formally self-dual binary codes of parameters [72,36,14][72,36,14], [56,28,12][56,28,12], [44,22,10][44,22,10] and odd formally self-dual binary codes of parameters [72,36,13][72,36,13], all of which have better minimum distances than the best known self-dual codes of the same lengths.
... Then [I 3 | M] generates a formally self-dual code of length 6 over F 5 + vF 5 + v 2 F 5 . The Gray image of this code is an [18,9,7] formally self-dual code over F 5 . ...
... generates a formally self-dual code of length 6 over F 3 + vF 3 + v 2 F 3 . The Gray image of this code is an [18,9,6] formally self-dual code over F 3 . This is an optimal code. ...
... We have that A = A t , so [I 3 | A] generates a formally self-dual code of length 6 over F 5 + vF 5 + v 2 F 5 . The Gray image of this code is an [18,9,7] formally self-dual code over F 5 . Example 4.9 Let q = 9, n = 5, and A be the matrix ...
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R=\F_{q}+v\F_{q}+v^{2}\F_{q}, where , for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \F_q and extend these to codes over R.
... Remark 1 Note that the formally self-dual codes constructed in Example 1 and Table 1 with parameters [40,20,9] Table 1 is not equivalent to the one in [9]. On the other hand, [72, 36, 14] 2 code in Table 1 is equivalent to the one in Table 2 of [9]. ...
... Remark 1 Note that the formally self-dual codes constructed in Example 1 and Table 1 with parameters [40,20,9] Table 1 is not equivalent to the one in [9]. On the other hand, [72, 36, 14] 2 code in Table 1 is equivalent to the one in Table 2 of [9]. ...
... Remark 1 Note that the formally self-dual codes constructed in Example 1 and Table 1 with parameters [40,20,9] Table 1 is not equivalent to the one in [9]. On the other hand, [72, 36, 14] 2 code in Table 1 is equivalent to the one in Table 2 of [9]. ...
We shall describe several families of X-rings and construct self-dual and formally self-dual codes over these rings. We then use a Gray map to construct binary formally self-dual codes from these codes. In several cases, we produce binary formally self-dual codes with larger minimum distances than known self-dual codes. We also produce non-linear codes which are better than the best known linear codes.
... We consider the duality relation using two different inner products. We apply the construction methods given in [12] to obtain self-dual codes over the ring R. We also analyze the binary images of self-dual codes over R. ...
... In this section we present two upper bounds and define the generator matrix of the Gray image of a free linear code over R. Later, we give the necessary and sufficient conditions in order to obtain free Euclidean and free Hermitian self-dual codes over R. At the end of this section, we recall three construction methods known as the Double Circulant Method (DCM), the Bordered Double Circulant Method (BDCM) and the Symmetric Method (SM) to find self-dual codes over R, which are proved in [12]. ...
In this study we consider Euclidean and Hermitian self-dual codes over the direct product ring where v2 = v. We obtain some theoretical outcomes about self-dual codes via the generator matrices of free linear codes over . Also, we obtain upper bounds on the minimum distance of linear codes for both the Lee distance and the Gray distance. Moreover, we find some free Euclidean and free Hermitian self-dual codes over via some useful construction methods.
... Constructing new self-dual codes and classifying self-dual codes have been an interesting research subject. Researchers have used different techniques to construct self-dual codes ( [9], [11], [14]). In the present paper, we deal with self-dual codes over the ring + with = of order 4. We use several known construction methods to establish free Euclidean self-dual codes. ...
... In Section 3, we study free Euclidean self-dual codes over the ring + using the Chinese remainder theorem. In Section 4, we find the highest possible minimum weights using the construction methods given in [14]. ...
In this paper, free Euclidean self-dual codes over the ring F2 + v F2 with v2 =v of order 4 are considered. A necessary and sufficient condition for the form of the generator matrix of a free Euclidean self-dual code is given. By using the distance preserving Gray map from F2 + v F2 to F2 x F2, the generator matrix of the binary code which corresponds the code over the ring F2 + v F2 is obtained. The codes of lengths up to 100 over the ring F2 + v F2 are found.
... In particular, binary self-dual codes have garnered extensive attention, with considerable research effort dedicated to developing techniques for constructing new extremal and optimal binary self-dual codes. These known construction techniques include the doublecirculant and bordered double-circulant constructions [1][2][3][4], the four-circulant construction and its variations [5][6][7][8][9][10][11], group rings and their connection to self-dual codes [12][13][14][15][16][17][18][19][20], bordered matrix constructions [12][13][14]17,19,[21][22][23][24], neighbours of binary self-dual codes [25,26], the widely employed building-up construction [27][28][29], the production of binary self-dual codes as Gray images of self-dual codes over finite commutative Frobenius rings of characteristic 2 [28,[30][31][32], and the well-known lifting method [10,11,[33][34][35][36][37]. ...
In this work, we present a new method for constructing self-dual codes over finite commutative rings R with characteristic 2. Our method involves searching for k×2k matrices M over R satisfying the conditions that its rows are linearly independent over R and MM⊤=α⊤α for an R-linearly independent vector α∈Rk. Let C be a linear code generated by such a matrix M. We prove that the dual code C⊥ of C is also a free linear code with dimension k, as well as C/Hull(C) and C⊥/Hull(C) are one-dimensional free R-modules, where Hull(C) represents the hull of C. Based on these facts, an isometry from Rx+Ry onto R2 is established, assuming that x+Hull(C) and y+Hull(C) are bases for C/Hull(C) and C⊥/Hull(C) over R, respectively. By utilizing this isometry, we introduce a new method for constructing self-dual codes from self-dual codes of length 2 over finite commutative rings with characteristic 2. To determine whether the matrix MM⊤ takes the form of α⊤α with α being a linearly independent vector in Rk, a necessary and sufficient condition is provided. Our method differs from the conventional approach, which requires the matrix M to satisfy MM⊤=0. The main advantage of our method is the ability to construct nonfree self-dual codes over finite commutative rings, a task that is typically unachievable using the conventional approach. Therefore, by combining our method with the conventional approach and selecting an appropriate matrix construction, it is possible to produce more self-dual codes, in contrast to using solely the conventional approach.
... In this section, we define a Gray map : B j,k → F 2 j +k p r . The map we give is a generalization of the map given in [1] as well as those given in [2,[6][7][8], and [9]. ...
In this work, we study a new family of rings, B_{j,k}, whose base field is the finite field F_{p^r}. We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over B_{j,k} to a code over B_{l,m} is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible G^{2j^{+k}}-code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also G^s-codes for some s.
... This has been successfully applied in works such as [10], [14], [15], [16], [17]. In [13], these ideas were applied for constructing formally self-dual codes of high minimum distances. ...
In this work, we introduce new construction methods for self-dual codes using a Baumert–Hall array. We apply the constructions over the alphabets and and combine them with extension theorems and neighboring constructions. As a result, we construct 46 new extremal binary self-dual codes of length 68, 26 new best known Type II codes of length 72 and 8 new extremal Type II codes of length 80 that lead to new designs. Among the new codes of length 68 are the examples of codes with the rare parameter in . All these new codes are tabulated in the paper.
... 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.
... This has been successfully applied in works such as [10], [14], [15], [16], [17]. In [13], these ideas were applied for constructing formally self-dual codes of high minimum distances. ...
In this work, we introduce new construction methods for self-dual codes using a Baumert-Hall array. We apply the constructions over the alphabets F_2 and F_4 + uF_4 and combine them with extension theorems and neighboring constructions. As a result, we construct 46 new extremal binary self-dual codes of length 68, 26 new best known Type II codes of length 72 and 8 new extremal Type II codes of length 80 that lead to new 3-(80,16,665) designs. Among the new codes of length 68 are the examples of codes with the rare \gamma= 5 parameter in W68;2. All these new codes are tabulated in the paper.
... , 1 + u k of the unit group of R k is called the subgroup of basic units. In [9], it is shown that multiplying by basic units corresponds to a permutation of coordinates in the Gray image, thus we have the following lemma: Lemma 3.6 (Lemma 2.1 in [12]). (1) The Lee weight of each basic unit is 1. ...
In this work, we consider constacyclic and cyclic self-dual codes over the rings Rk. We start with theoretical existence results for constacyclic and cyclic self-dual codes of any length over Rk and then construct cyclic self-dual codes over R1 = F2 + uF2 of even lengths from lifts of binary cyclic self-dual codes. We classify all free cyclic self-dual codes over R1 of even lengths for which non-trivial such codes exist. In particular we demonstrate that our constructions provide a counter example to a claim made by Batoul et al. in [1] and we explain why their claim fails.
... Let G be the cyclic group of order 10 and v = 1+uh+h 5 +uh 9 ∈ R 1 C 10 . Then C v = σ(v), uσ(v) is cyclic self-dual code and its image under φ 1 is a binary quasi-cyclic self-dual [20,10,4] code of index 2. ...
We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in \cite{Hurley1} by showing that one of the conditions given in the theorem is unnecessary and moreover it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes correspond to ideals in the group ring RG and as such must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72,36,16] Type~II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48.
... 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 [1,2,9]), Z 4 [x]/ x 2 (see [10]) 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.
... Especially the binary near extremal formally self-dual (f.s.d.) codes have been studied extensively. For more detail we refer the reader to [4,7,8] which have better distances than the best known self-dual codes of the corresponding lengths. ...
In this paper, we investigate the structure and properties of duadic, isodual cyclic and formally
self-dual codes over the ring R = F_q + vF_q with v2 = v. In addition to the theoretical work on
the structure of these codes, we construct examples of good codes over different alphabets from cyclic
self-dual and formally self-dual codes over R.
... Estos anillos pueden ser fácilmente caracterizados como aquellos que tienen un socle simple. Varios ejemplos denominados clases R k and S k y las aplicaciones de Gray relacionadas con ellas han sido propuestos en los trabajos de Dougherty et al. (véase por ejemplo [16,17,20]). También recientemente se ha realizado una primera clasificación de los anillos finitos con la condición de Frobenius locales con 16 elementos y sus posibles aplicaciones de Gray asociadas [25]. ...
We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in \cite{Hurley1} by showing that one of the conditions given in the theorem is unnecessary and moreover it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes correspond to ideals in the group ring RG and as such must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72,36,16] Type~II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48.
In this paper, we investigate free Hermitian self-dual codes whose generator matrices are of the form [I,A+vB] over the ring F2+vF2={0,1,v,1+v} with v2=v. We use the double-circulant, the bordered double-circulant and the symmetric construction methods to obtain free Hermitian self-dual codes of even length. By describing a new shortening method over this ring, we are able to obtain Hermitian self-dual codes of odd length. Using these methods, we also obtain a number of extremal codes. We tabulate the Hermitian self-dual codes with the highest minimum weights of lengths up to 50.
Let Rk,m be the ring 2m[u1,⋯,uk] (ui2,uiuj-ujui). We study cyclic codes of arbitrary length n over the ring R3,m and obtain a complete characterization for them by presenting a unique set of generators for these codes. For example, we try to determine all distinct cyclic codes of length 2 over the ring R3,m by a practical algorithm.
Let R be the finite chain ring R=Fpm[u]∕〈u⁴〉 where p is a prime and m is a positive integer. In this work, we give a complete classification of (1+αu²)-constacyclic codes of length pk over R, where α is a nonzero element of Fpm. We also completely determine self-dual such codes and enumerate them. Finally we discuss on Gray-maps on R which preserve self-duality, and also discuss on the images of self-dual constacyclic codes under these Gray maps.
We present three explicit methods for construction of formally self-dual codes over ℤ
4
. We characterize relations between Lee weight enumerators of formally self-dual codes of length n over ℤ
4
and those of length n t 2; the first two construction methods are based on these relations. The last construction produces free formally self-dual codes over ℤ
4
. Using these three constructions, we can find free formally selfdual codes over ℤ
4
, as well as non-free formally self-dual codes over ℤ
4
of all even lengths. We find free or non-free formally selfdual codes over ℤ
4
of lengths up to ten using our constructions. In fact, we obtain 46 inequivalent formally self-dual codes whose minimum Lee weights are larger than self-dual codes of the same length. Furthermore, we find 19 non-linear extremal binary formally self-dual codes of lengths 12, 16, and 20, up to equivalence, from formally self-dual codes over ℤ
4
by using the Gray map.
In this chapter, we describe self-dual codes over Frobenius rings. We give constructions of self-dual codes over any Frobenius ring. We describe connections to unimodular lattices, binary self-dual codes and to designs. We also describe linear complementary dual codes and make a new definition of a broad generalization encompassing both self-dual and linear complementary codes.
In this chapter, we describe families of rings including the rings of order 4, their generalizations, X-rings, and the ring . We describe the kernel and rank of binary codes that are images of quaternary codes via the Gray map. Then a generalized Singleton bound is proven for codes over Frobenius rings.
We define a class of finite Frobenius rings of order , describe their generating characters, and study codes over these rings. We define two conjugate weight preserving Gray maps to the binary space and study the images of linear codes under these maps. This structure couches existing Gray maps, which are a foundational idea in codes over rings, in a unified structure and produces new infinite classes of rings with a Gray map. The existence of self-dual and formally self-dual codes is determined and the binary images of these codes are studied.
In this paper, we study the linear codes over the commutative ring
R=\F_{q}+v\F_{q}+v^{2}\F_{q} and their Gray images, where . We
define the Lee weight of the elements of R, we give a Gray map from
to \F^{3n}_{q} and we give the relation between the dual and the Gray image
of a code. This allows us to investigate the structure and properties of
self-dual cyclic, formally self-dual and the Gray image of formally self-dual
codes over R. Further, we give several constructions of formally self-dual
codes over $R
A code C is called formally self-dual if C and C? have the same weight enumerators. There are four types of nontrivial divisible formally self-dual codes overF2,F3, andF4. These codes are called extremal if their minimum distances achieve the Mallows-Sloane bound. S. Zhang gave possible lengths for which extremal self-dual codes do not exist. In this paper, we deflne near-extremal formally self-dual (f.s.d.) codes. With Zhang's systematic approach, we determine possible lengths for which the four types of near- extremal formally self-dual codes as well as the two types of near-extremal formally self-dual additive codes cannot exist. In particular, our result on the nonexistence of near-extremal binary f.s.d. even codes of any even length n completes all the cases since only the case 8jn was dealt with by Han and Lee.
Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.
. 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 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.
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.
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.
Bachoc bachoc has recently introduced harmonic polynomials for binary codes. Computing these for extremal even formally self-dual codes of length 12, she found intersection numbers for such codes and showed that there are exactly three inequivalent [12,6,4] even formally self-dual codes, exactly one of which is self-dual. We prove a new theorem which gives a generator matrix for formally self-dual codes. Using the Bachoc polynomials we can obtain the intersection numbers for extremal even formally self-dual codes of length 14. These same numbers can also be obtained from the generator matrix. We show that there are precisely ten inequivalent [14,7,4] even formally self-dual codes, only one of which is self-dual.
In this paper we prove that there are exactly 7 inequivalent [20,10,6] even extremal formally self-dual binary codes and that there are over 1000 inequivalent [22,11,6] even extremal formally self-dual binary codes. We give properties of these codes and present a summary of what is known about the classification of even extremal formally self-dual codes of small length.
A binary code with the same weight distribution as its dual code is called formally self-dual ( f.s.d.). We only consider f.s.d. even codes (codes with only even weight codewords). We show that any formally self-dual even binary code C of length n not divisible by 8 is balanced. We show that the weight distribution of a balanced near-extremal f.s.d. even code of length a multiple of 8 is unique. We also determine the possible weight enumerators of a near-extremal f.s.d. even [n,n/2,2⌊n/8⌋] code with 8|n as well as the dimension of its radical.
It is a well-known fact that if C is an [n,k,d] formally self-dual even code with n>30, then d⩽2[n/8]. A formally self-dual (f.s.d.) even code with d=2[n/8] is called near-extremal. Kim and Pless [A note on formally self-dual even codes of length divisible by 8, Finite Fields Appl., available online 13 October 2005.] conjecture that there does not exist a near-extremal f.s.d. (not Type II) code of length n⩾48 with 8|n. In this paper, we prove that if n⩾72 and 8|n, then there is no near-extremal f.s.d. even code. This result comes from the negative coefficients of weight enumerators. In addition, we introduce shadow transform in near-extremal f.s.d. even codes. Using this we present some results about the nonexistence of near-extremal f.s.d. even codes with n=48,64.
Most of the research on formally self-dual (f.s.d.) codes has been developed for binary f.s.d. even codes, but only limited research has been done for binary f.s.d.
odd codes. In this article we complete the classification of binary f.s.d. odd codes of lengths up to 14. We also classify
optimal binary f.s.d. odd codes of length 18 and 24, so our result completes the classification of binary optimal f.s.d. odd
codes of lengths up to 26. For this classification we first find a relation between binary f.s.d. odd codes and binary f.s.d.
even codes, and then we use this relation and the known classification results on binary f.s.d. even codes. We also classify
(possibly) optimal binary double circulant f.s.d. odd codes of lengths up to 40.
Binary formally self-dual (f.s.d.) even codes are the one type of divisible [2n, n] codes which need not be self-dual. We examine such codes in this paper. On occasion a f.s.d. even [2n, n] code can have a larger minimum distance than a [2n, n] self-dual code. We give many examples of interesting f.s.d even codes. We also obtain a strengthening of the Assmus-Mattson theore. IfC is a f.s.d. extremal code of lengthn≡2 (mol 8) [n ≡6 (mod 8)], then the words of a fixed weight inC ∪C
⊥ hold a 3-design [1-design]. Finally, we show that the extremal f.s.d. codes of lengths 10 and 18 are unique.
In the rst of two papers on Magma, a new system for computational algebra, we present the Magma language, outline the design principles and theoretical background, and indicate its scope and use. Particular attention is given to the constructors for structures, maps, and sets. c 1997 Academic Press Limited Magma is a new software system for computational algebra, the design of which is based on the twin concepts of algebraic structure and morphism. The design is intended to provide a mathematically rigorous environment for computing with algebraic struc- tures (groups, rings, elds, modules and algebras), geometric structures (varieties, special curves) and combinatorial structures (graphs, designs and codes). The philosophy underlying the design of Magma is based on concepts from Universal Algebra and Category Theory. Key ideas from these two areas provide the basis for a gen- eral scheme for the specication and representation of mathematical structures. The user language includes three important groups of constructors that realize the philosophy in syntactic terms: structure constructors, map constructors and set constructors. The util- ity of Magma as a mathematical tool derives from the combination of its language with an extensive kernel of highly ecient C implementations of the fundamental algorithms for most branches of computational algebra. In this paper we outline the philosophy of the Magma design and show how it may be used to develop an algebraic programming paradigm for language design. In a second paper we will show how our design philoso- phy allows us to realize natural computational \environments" for dierent branches of algebra. An early discussion of the design of Magma may be found in Butler and Cannon (1989, 1990). A terse overview of the language together with a discussion of some of the implementation issues may be found in Bosma et al. (1994).
We introduce a class of formally self-dual additive codes over F4 as a natural ana- logue of binary formally self-dual codes, which is missing in the study of additive codes over F4. We deflne extremal formally self-dual additive codes over F4 and classify all such codes. Interestingly, we flnd exactly three formally self-dual additive (7;2 7 ) odd codes overF4 with minimum distance d = 4, a better minimum distance than any self- dual additive (7;2 7 ) codes overF4. We further deflne near-extremal formally self-dual additive codes overF4 as an analogue of near-extremal binary formally self-dual codes and prove that they do not exist if their lengths are n = 16;18 or n ‚ 20.
Bound on the minimum distance of linear codes and quantum codes. Online available at: www.codetables.de (accessed on 07
- M Grassl
M. Grassl, Bound on the minimum distance of linear codes and quantum codes. Online available at: www.codetables.de (accessed on 07.2012).