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Coding theory and the Mathieu groups
Abstract
Let ℳ denote the Mathieu group on 24 points. Let G be the subgroup of ℳ which has three sets of transitivity, the eight points on a Golay code-word (or Steiner octuple), one additional point, and the remaining 15 points. Using elementary results from the subject of algebraic coding theory, we present a new proof of the fact that G acts on the eight points as the alternating group, A8, and on the 15 points as the general linear group, G ℳ(4, 2). This result and other properties of the Mathieu groups obtained from it are then used to obtain the symmetry groups of the Nordstrom—Robinson nonlinear (15, 8) code and the linear, cyclic (15, 7) and (21, 12) BCH codes and the (21, 10) dual of a projective geometry code, all of which have distance 5.
... Berlekamp proved that N is a binary (16, 256, 6) code, and that Aut(N ) 0 = Perm(N ) = 2 4 : A 7 acting 3-transitively on 16 points [3], where 0 is the zero codeword in N . We also require the following. ...
... The Nordstrom-Robinson code is the first member the Preparata codes [35], an infinite family of non-linear binary codes. For each odd k ≥ 3, the Preparata code P(k) has length 2 k+1 , contains 2 k+1 − 2(k + 1) codewords and has minimum distance 6 (see for example, [29, Section 7.4.3]). The code P(3) is equivalent to the Nordstrom-Robinson code N of length 16. ...
... The punctured Nordstrom-Robinson code PN is a (15, 256, 5) code (see, for example, [35]). Moreover, since all (15, 256, 5) codes are equivalent, we can assume without loss of generality that PN is obtained from N by puncturing the first entry, as in [3]. Recall also (Remark 2) that PN has covering radius 3. ...
In his doctoral thesis, Snover proved that any binary (m,256,δ) code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for (m,δ)=(16,6) or (15,5), respectively. We prove that these codes are also characterised as completely regular binary codes with (m,δ)=(16,6) or (15,5), and moreover, that they are completely transitive. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular codes are not completely transitive, namely the (punctured) Preparata codes other than the (punctured) Nordstrom-Robinson code.
... Its existence depends on an easy result from group extension theory. The Optimism Code has isometry group a nonsplit extension 2 1+8 GL (4,2) and is connected to a nonlinear binary Nordstrom-Robinson type code. For the latter code, we have an easy existence proof and determination of its automorphism group. ...
... Remark 2. 4. The defect of an involution t ∈ G is the integer k so that ...
... (ii) The first isomorphism is realized by the action of p 01 . For the second, note that T 1 may be identified with linear functionals on Ω which have 4. We continue to use the notations of (4.1). ...
Using Barnes–Wall lattices and 1-cocycles on finite groups of monomial matrices, we give a procedure to construct tricosine spherical codes. This was inspired by a 14-dimensional code which Ballinger, Cohn, Giansiracusa and Morris discovered in studies of the universally optimal property. Their code has 64 vectors and cosines . We construct the Optimism Code, a 4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are . Its automorphism group has shape 21+8⋅GL(4,2). The Optimism Code contains a subcode related to the BCGM code. The Optimism Code implies existence of a nonlinear binary code with parameters (16,256,6), a Nordstrom–Robinson code, and gives a context for determining its automorphism group, which has form .
... Le résultat de * s'appelle syndrome, les algorithmes de décodage des codes en bloc sont généralement basés sur le calcul du syndrome. les décodages par syndrome les plus connus sont l'algorithme d'Euclide [79] ainsi que l'algorithme de Berlekamp Massey [80] [81]. Cependant dans le cas des codes LDPC, on utilise un algorithme de décodage itératif basé sur le critère du maximum vraisemblance. ...
... L'algorithme le plus répandu pour le décodage RS est l'algorithme de Berlekamp-Massey [81][86] [80]. Il est basé sur des décisions dures. ...
De nos jours, l’architecture du réseau mobile est en pleine évolution pour assurer la montée en débit entre les Centraux (CO) (réseaux coeurs) et différents terminaux comme les mobiles, ordinateurs, tablettes afin de satisfaire les utilisateurs. Pour faire face à ces défis du futur, le réseau C-RAN (Cloud ou Centralized-RAN) est connu comme une solution de la 5G. Dans le contexte C-RAN, toutes les BBUs (Base Band Units) sont centralisées dans le CO, seules les RRH (Remote Radio Head) restent situées à la tête de la station de base (BS). Un nouveau segment entre les BBUs et RRHs apparait nommé « fronthaul ». Il est basé sur des transmissions D-ROF (digital radio-overfiber) et transporte le signal radio numérique à un débit binaire élevé en utilisant le protocole CPRI (Common Public Radio Interface). En prenant en compte le CAPEX et l’OPEX, le projet ANR LAMPION a proposé la technologie RSOA (Reflective Semiconductor Optical Amplifier) auto alimenté afin de rendre la solution plus flexible et s’affranchir d’émetteurs/récepteurs colorés dans le cadre de transmission WDM-PON (Wavelength Division Multiplexing Passive Optical Network). Néanmoins, il est nécessaire d’ajouter un FEC (forward error corrector) dans la transmission pour assurer la qualité de service. Donc l’objectif de cette thèse est de trouver le FEC le plus adéquat à appliquer dans le contexte C-RAN. Nos travaux se sont focalisés sur l’utilisation de codes LDPC, choisis après comparaisons des performances avec les autres types de codes. Nous avons précisé les paramètres (rendement du code, taille de la matrice, cycle, etc.) nécessaires pour les codes LDPC afin d'obtenir les meilleures performances. Les algorithmes LDPC à décisions dures ont été choisis après considération du compromis entre complexités de circuit et performance. Parmi ces algorithmes à décision dures, le GDBF (gradient descent bit-flipping) était la meilleure solution. La prise en compte d’un CAN 2-Bit dans le canal nous a amené à proposer une variante : le BWGDBF (Balanced weighted GDBF). Des optimisations ont également été faites en regard de la convergence de l'algorithme et de la latence. Enfin, nous avons réussi à implémenter notre propre algorithme sur le FPGA Spartan 6 xc6slx16. Plusieurs méthodes ont été proposées pour atteindre une latence de 5 μs souhaitée dans le contexte C-RAN. Cette thèse a été soutenue par le projet ANR LAMPION (Lambada-based Access and Metropolitan Passive Optical networks).
... This element is called coset leader. It is known [6] that this code is formed of eight cosets of the Reed-Muller code of order 1 with parameters (16, 5,8). ...
... To be precise, if we rename the bits Y 0 , ..., Y 7 as X 8 , ..., X 15 and set Z i = X π(i) where π is the following order 14 cycle permutation: π = (2, 10,13,11,16,5,9,4,12,8,6,7,15,14) ( ...
The Nordstrom-Robinson code is a binary nonlinear code of length 16 with 28 = 256 codewords. Its minimum distance which is equal to 6 is greater than that of any binary linear code of length 16 and encoding rate 1/2. For this reason, the Nordstrom-Robinson code has attracted interests in practical applications of the communication area. In this paper, we present two new algorithms for soft decoding of this code.
... It is easy to see that if C = φ(C) is the binary image of a linear quaternary code C, then Aut(C) is isomorphic to a subgroup of Aut(C). The automorphism groups of the binary Nordstrom-Robinson, Kerdock, classical Preparata, and Delsarte-Goethals codes are known (Berlekamp [3], Carlet [14], [15], [16], Kantor [44, 45]). ...
... The case m = 3 is exceptional. The quaternary octacode has an automorphism group of order 1344 (Conway and Sloane [23]), whereas the group of the binary Nordstrom-Robinson code has order 80640 (Berlekamp [3], see also Conway and Sloane [21]). 5.6. ...
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.
... Berlekamp proved that N is a binary (16, 256, 6) code, and that Perm(N ) = 2 4 : A 7 acting 3 transitively on 16 points [2]. Therefore, by our definition of the automorphism group of a code, Aut(N ) 0 = 2 4 : A 7 , where 0 is the zero codeword in N . ...
... Thus, for any p ′ ∈ M , the punctured N code with respect to p ′ is equivalent to π J (N ). Therefore, without loss of generality, we can assume that p = 1 as in [2], and we denote π J (N ) by PN . By [2, Lemma 6.5], Aut(PN ) 0 ∼ = A 7 acting 2 -transitively on 15 points. ...
In his doctorate thesis, Snover proved that any binary (m,256,\delta) code is
equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson
code for (m,\delta)=(16,6) or (15,6) respectively. By replacing the condition
that the code consists of 256 codewords with the requirement that the code is
completely regular, we prove that the same result holds. Moreover, we prove
that these codes are completely transitive.
... In 1973 Sloane [34] posed a question which remains unresolved: is there a binary self-dual doubly-even [72,36,16] code? The automorphism group of the extended Golay code is the 5-transitive Mathieu group M 24 of order 2 10 ·3 3 ·5·7·11·23 (see [3]), as the automorphism group of q 48 is only 2-transitive and is isomorphic to the projective special linear group PSL(2, 47) of order 2 4 · 3 · 23 · 47 [25]. The first authors to study the automorphism group of the putative [72, 26,16] code were Conway and Pless [13], in particular they focused on the possible automorphisms of odd prime order. ...
... In 1973 Sloane [34] posed a question which remains unresolved: is there a binary self-dual doubly-even [72, 36, 16] code? The automorphism group of the extended Golay code is the 5-transitive Mathieu group M 24 of order 2 10 ·3 3 ·5·7·11·23 (see [3]), as the automorphism group of q 48 is only 2-transitive and is isomorphic to the projective special linear group PSL(2, 47) of order 2 4 · 3 · 23 · 47 [25]. The first authors to study the automorphism group of the putative [72, 26,16] code were Conway and Pless [13], in particular they focused on the possible automorphisms of odd prime order. ...
The purpose of this paper is to present the structure of the linear codes over a finite field with q elements that have a permutation automorphism of order m. These codes can be considered as generalized quasi-cyclic codes. Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail, presenting necessary and sufficient conditions for which linear codes with such an automorphism are self-orthogonal, self-dual, or linear complementary dual.
... The other code meets |Sym(C)| = 3|Sym π (C)|. The order of the symmetry group of the unique Preparata-like code of length 16 is 16 · 15 · 14 · 12 [2], see also [4]. ...
We consider the symmetry group of a -linear code with parameters of a 1-perfect, extended 1-perfect, or Preparata-like code. We show that, provided the code length is greater than 16, this group consists only of symmetries that preserve the structure. We find the orders of the symmetry groups of the -linear (extended) 1-perfect codes. Keywords: additive codes, -linear codes, 1-perfect codes, Preparata-like codes, automorphism group, symmetry group.
... The codewords of weight 8 form the octuples of a Steiner system 5(5, 8, 24). For later use we observe that if the eight coordinates belonging to an octuple are deleted from all the codewords, the truncated codewords form two copies of a (16, 2 11 , 4) 2nd order RM code (see [5]). Other extended quadratic residue codes are the (12, 3 6 , 6) ternary Golay code (see [13; 4, p. 359]), and the (24, 3 12 , 9) and (48, 3 24 , 15) ternary codes studied by Assmus and Mattson [1; 2]. ...
Error-correcting codes are used in several constructions for packings of equal spheres in n -dimensional Euclidean spaces E ⁿ . These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2 m for m ≧ 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density Δ of the packings in E ⁿ for n = 2 m satisfies
In this paper we make systematic use of error-correcting codes to obtain sphere packings in E ⁿ , including several of the densest packings known and several new packings.
... As it was noticed above, the Nordstrom–Robinson code of length 16 is transitive. In [18] it is proved that the group of symmetries of this code is three times transitive. Therefore by Lemma 4 the code of length 15 obtained from the Nordstrom–Robinson code by a puncturing any coordinate is transitive. ...
It is shown that among all Preparata codes only the code of length 16 is distance regular. An analogous result takes place
for Preparata codes after puncturing any coordinate (only the code of length 15 is distance regular).
All simple finite groups are classified as members of specific families. With one exception, these families are infinite collections of groups sharing similar structures. The exceptional family of sporadic groups contains exactly twenty-six members. The five Mathieu groups are the most accessible of these sporadic cases. In this article, we explore connections between Mathieu groups and error-correcting communication codes. These connections permit simple, visual representations of the three largest Mathieu groups: M24, M23, and M22. Along the way, we provide a brief, nontechnical introduction to the field of coding theory.
In this chapter we describe one of the most important of all codes, the [24,12,8] extended Golay code.
A code is said to be propelinear if its automorphism group contains a subgroup which acts on the codewords regularly. Such a subgroup is called a propelinear structure on the code. With the aid of computer, we enumerate all propelinear structures on the Nordstrom–Robinson code and analyze the problem of extending them to propelinear structures on the extended Hamming code of length 16. The latter result is based on the description of partitions of the Hamming code of length 16 into Nordstrom–Robinson codes via Fano planes, presented in the paper. As a result, we obtain a record-breaking number of propelinear structures in the class of extended perfect codes of length 16.
Algebraic geometry is often employed to encode and decode signals transmitted in communication systems. This book describes the fundamental principles of algebraic coding theory from the perspective of an engineer, discussing a number of applications in communications and signal processing. The principal concept is that of using algebraic curves over finite fields to construct error-correcting codes. The most recent developments are presented including the theory of codes on curves, without the use of detailed mathematics, substituting the intense theory of algebraic geometry with Fourier transform where possible. The author describes the codes and corresponding decoding algorithms in a manner that allows the reader to evaluate these codes against practical applications, or to help with the design of encoders and decoders. This book is relevant to practicing communication engineers and those involved in the design of new communication systems, as well as graduate students and researchers in electrical engineering.
We prove that the automorphism group of the Kerdock code of length 2 m (m even ≥6) is the group of permutations on GF(2 m )≈GF(2 m-1 )×GF(2) generated by the automorphism group of the field GF(2 m-1 ), and the translations on GF(2).
Although more than twenty years have passed since the appearance of Shannon's papers, a still unsolved problem of coding theory is to construct block codes which attain a low probability of error at rates close to capacity. However, for moderate block lenghts many good codes are known, the best-known being the BCH codes discovered in 1959. This paper is a survey of results in coding theory obtained since the appearance of Berlekamp's “Algebraic coding theory” (1968), concentrating on those which lead to the construction of new codes. The paper concludes with a table giving the smallest redundancy of any binary code, linear or nonlinear, that is presently known (to the author), for all lengths up to 512 and all minimum distances up to 30.
Two earlier versions of our bibliography on Steiner systems [D44], [D44.1] listed 400 and 675 titles, respectively. The number of papers concerning Steiner systems continues to grow, and the present listing contains already some 730 titles; all additions are recent papers. As for inclusion into our bibliography of papers dealing with finite projective, affine or inversive spaces, we follow the same policy as in [D44] and [D44.1]. Information about reviews in referative journals is attached to the titles in the bibliography; MR stands for Mathematical Reviews while Z stands for Zentral-blatt für Mathematik und ihre Grenzgebiete.
This chapter provides an overview of symmetric Hadamard matrix of order 36. A square matrix H is a Hadamard matrix if its elements are + 1 and if it is orthogonal. If in addition, H is symmetric, has a constant diagonal I, and has order 36, then H2 = 36I, H = HT, H = A + I, (A−5I) (A+7I) = 0.The chapter explains a method to obtain symmetric Hadamard matrices of order 36 from the 12 Latin squares of order 6, and from the 80 Steiner triple systems of order 15 . The graphs obtained in this way are shown to be nonequivalent with respect to switching, with one exceptional pair. The method consists of counting void and complete sub-graphs, and is capable of wider application. The chapter also explains how graphs of the Steiner type can be switched into strongly regular graphs. Among these graphs, there are 23 that can be made regular with valency 21, whereas all 80 can be made regular with valency I5. This follows from a computer search by which each of the 80 Steiner triple systems appears to contain a 3-factor, that is, a subsystem of 15 triples in which every symbol occurs exactly three times.
We show that the Nordstrom-Robinson (1968) code may be represented
as the union of binary images of two isomorphic linear (4,2,3) codes
over GF(4). Certain symmetries of the unique quaternary (4,2,3)
quadracode are discussed. It is shown how the properties of the
Nordstrom-Robinson code itself, such as weight distribution, distance
invariance, and the well-known representation as the union of eight
cosets of the (16,5,8) Reed-Muller code, may be easily rederived from
these symmetries. In addition we introduce a decoding algorithm for the
Nordstrom-Robinson code which involves projecting its codewords onto the
codewords of the quadracode. The algorithm is simple enough to enable
hard-decision decoding of the Nordstrom-Robinson code by hand.
Furthermore we present an algorithm for maximum-likelihood soft-decision
decoding of the Nordstrom-Robinson code based on its representation over
GF(4). The complexity of the proposed algorithm is at most 205 real
operations. This is, to the best of our knowledge, less than the
complexity of any existing decoder, and twice as efficient as the fast
maximum-likelihood decoder of Adoul (1987). We also investigate several
related topics, such as bounded-distance decoding, sphere-packings, and
the (20,2048,6) code
We determine the automorphism groups of the Delsarte-Goethals codesDG(m, d) (m=2t+2=6, 2=d=t). The groups that we obtain are the same as those of the Kerdock codesK(m) of the same length.
This paper introduces a new class of nonlinear binary codes. For each l = 2, 3,… we present a code with 241 codewords of length N = 4l and distance d = (4l — 2l)/2. Each code is a supercode of the 1st-order Reed-Muller (RM) code and a subcode of the 2nd-order RM code. These codes are the “duals≓ of the extended nonlinear Preparata codes in the sense that their weight distributions satisfy the MacWilliams identities.
New elementary proofs of the uniqueness of certain Steiner systems using coding theory are presented. In the process some of the codes involved are shown to be unique.The uniqueness proof for the (5, 8, 24) Steiner system is due to John Conway. The blocks of the system are used to generate a length 24 binary code. Any two such codes are then shown to be equivalent up to a permutation of the coordinates. This code turns out to be the extended Golay code.In the uniqueness proof for the (4, 7, 23) system, the blocks generate a length 23 code which is extended to a length 24 code. The minimum weight vectors of this larger code hold a (5, 8, 24) Steiner system. This result together with the previous one completes the proof. At this point it is also possible to conclude that the codes involved are unique and hence equivalent to the binary perfect Golay code and its extension.Continuing with the uniqueness result for the (3, 6, 22) Steiner system, the blocks generate a length 22 code which is extended to the same length 24 code by the addition of two coordinates and one additional vector. This extension ultimately requires the computation of the coset weight distribution of the length 22 code, a result heretofore unknown. The complete coset weight distribution for a specific (22, 11, 6) self-dual code is computed using the CAMAC computer system.The (5, 6, 12) and (4, 5, 11) Steiner systems are treated differently. It is shown that each system is completely determined by the choice of six blocks which may be assumed to lie in any such design. These six blocks in fact form a basis for length 12 (and 11) ternary codes corresponding to the two systems and may be generated by an algorithm independent of the designs. This algorithm is presented and the minimum weight vectors of the resulting codes, the perfect ternary Golay code and its extension, are calculated by the CAMAC system.
A complete classification is given of all [22, 11] and [24, 12] binary self-dual codes. For each code we give the order of its group, the number of codes equivalent to it, and its weight distribution. There is a unique [24, 12, 6] self-dual code. Several theorems on the enumeration of self-dual codes are used, including formulas for the number of such codes with minimum distance ⩾ 4, and for the sum of the weight enumerators of all such codes of length n. Selforthogonal codes which are generated by code words of weight 4 are completely characterized.
In this correspondence, we prove that the automorphism group of a putative binary self-dual doubly even [72,36,16] code is solvable. Moreover, its order is 5,7,10,14,56, or a divisor of 72
The extended Nordstrom-Robinson code is an optimum nonlinear double-error-correcting code of length 16 with considerable practical importance. This code is also useful as a rate 1/2 vector quantizer for random waveforms such as speech linear-predictive-coding (LPC) residual. A fast decoding algorithm is described for maximum likelihood decoding (or nearest neighbor search in the squared-error sense) with 304 additions and 128 comparisons.
If m is odd and sigma /in Aut GF(2^{m}) is such that x rightarrow x^{sigma^{2}-1} is 1-1 , there is a [2^{m+1}-1,2^{m+l}-2m-2] nonlinear binary code P(sigma) having minimum distance 5. All the codes P(sigma) have the same distance and weight enumerators as the usual Preparata codes (which rise as P(sigma) when x^{sigma}=x^{2}) . It is shown that P(sigma) and P(tau) are equivalent if and only if tau=sigma^{pm 1} , and Aut P(sigma) is determined.
Constructions for a 32-word binary code of length 12 and minimum distance 5 were given by Nadler in 1962 and van Lint in 1972. These codes are not equivalent as their distance distributions are not the same. That their extended codes of length 13 are equivalent is proved. It is shown that up to a permutation of the coordinates, there is an essentially unique way to construct the extended code. This unique extended code contains only two inequivalent punctured codes of length 12.
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
A mixed code of covering radius 1 that has 60 codewords is
constructed. This code is then used to show that
K (10,1)⩽120
Zero-concurring codewords disclose a certain structure of the code
that may be used for efficient soft-decision decoding and for designing
DC-free codes. Methods for constructing sets of zero-concurring
codewords are presented for several families of codes. For the general
case an algorithm solution of the problem is offered. A table of results
obtained using the proposed techniques is supplied for all the primitive
narrow-sense binary BCH codes of length up to 127
These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently.
The principal results of these notes are the following. New sphere packings are given in [2 m ], m ⩾ 6, and in [24], which are twice as dense as those of §§1.6, 2.3. Others are given in [2 m ], m ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.
This paper is concerned with the packing of equal spheres in Euclidean spaces [ n ] of n > 8 dimensions. To be precise, a packing is a distribution of spheres any two of which have at most a point of contact in common. If the centres of the spheres form a lattice, the packing is said to be a lattice packing . The densest lattice packings are known for spaces of up to eight dimensions (1, 2) , but not for any space of more than eight dimensions. Further, although non-lattice packings are known in [3] and [5] which have the same density as the densest lattice packings, none is known which has greater density than the densest lattice packings in any space of up to eight dimensions, neither, for any space of more than two dimensions, has it been shown that they do not exist.
Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.
In § 2, a fibre-type construction for graphs is given which, applied to block designs with λ = 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande [22], in which the number of points in the intersection of any pair of blocks attains only two values.
A systematic nonlinear code having length 15, minimum distance 5, and 256 code words is given in Boolean form. This is the maximum possible number of words for length 15 and distance 5. The distance spectrum of all pairs of code words is an exact multiple of the weight spectrum of the code words.
The Mathieu group M24 can be represented as a collineation group in space of 11 dimensions over the field of two elements. This paper discusses
the geometry associated with this representation.
The research reported in this paper was supported by the National Science Foundation under the Research Participation Program for Exceptional Secondary Students and Grant GK-816.
Some combinatorial properties of the Golay binary code are emphasized and used for two main purposes. First, a new nonlinear code having 256 code words of length 16 at mutual distance 6 is exhibited. Second, a majority decoding method, quite similar to Massey's threshold decoding, is devised for the Golay code and various related codes.
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. This thesis deals with the problem of how the elements from a finite field F of characteristic p are distributed among the various linear recurrent sequences with a given fixed characteristic polynomial [...]. The first main result is a method of extending the so-called "classical method" for solving linear recurrences in terms of the roots of f. The main difficulty is that f might have a root [...] which occurs with multiplicity exceeding p-1; this is overcome by replacing the solutions [...], [...], [...], ..., by the solutions [...], [...], [...], .... The other main result deals with the number N of times a given element [...] appears in a period of the sequence, and for [...], the result is of the form [...] where [...] is an integer which depends upon f, but not upon the particular sequence in question. Several applications of the main results are given.
A group of order 8, 315, 553, 613, 086, 720
- Conway
GOLAY is the augmented quadratic residue code of length 23 over GF(2) XGOLAY is the extended quadratic residue code of length 24 over GF(2). SPC8 is the single parity check code of length 8 over OF(2), whose 2 7 codewords consist of all vectors of length 8 whose weight is even
- Elementary Properties Of The Binary Golay Codes
- Definitions
ELEMENTARY PROPERTIES OF THE BINARY GOLAY CODES DEFINITIONS 4.1 [pp. 352--353]. GOLAY is the augmented quadratic residue code of length 23 over GF(2). XGOLAY is the extended quadratic residue code of length 24 over GF(2). SPC8 is the single parity check code of length 8 over OF(2), whose 2 7 codewords consist of all vectors of length 8 whose weight is even. LEMMA 4.2 [Theorem 15.25 and 15.26, pp. 357-359]. XGOLAY is invariant under a transitive permutation group, and therefore [l GOLAY 11 II XGOLAY Ii --1 and II GOLAY II is odd. LEMMA 4.
- E F Jr
- H F And Mattson
ASSMUS, E. F., Jr., AND MATTSON, H. F. (1966), Perfect codes and Mathieu groups,
Arch. Math. 17, 121-135.