A characterization of 1-perfect additive codes (original) (raw)

On perfect codes and related concepts

Designs, Codes and …, 2001

The concept of diameter perfect codes, which seems to be a natural generalization of perfect codes (codes attaining the sphere-packing bound) is introduced. This was motivated by the "code-anticode" bound of Delsarte in distance regular graphs. This bound in conjunction with the recent complete solutions of diametric problems in the Hamming graph H q (n) and the Johnson graph J (n, k) gives a sharpening of the sphere-packing bound. Some necessary conditions for the existence of diameter perfect codes are given. In the Hamming graph all diameter perfect codes over alphabets of prime power size are characterized. The problem of tiling of the vertex set of J (n, k) with caps (and maximal anticodes) is also examined.

On traceability property of equidistant codes

Discrete Mathematics, 2017

Necessary and Sufficient conditions for an equidistant code to be a 2-TA code are obtained. An explicit construction method is proposed to obtain linear MDS [p + 1, 2, p] codes over the finite field F p , where p is a prime. These codes can be used as 2-TA codes for p > 2. In particular, for p = 3, it is observed that the linear [4, 2, 3] MDS code contradicts a result of Jin and Blaum (2007). The correct version of this result and its proof is given. Existence of some infinite families of equidistant 2-TA codes is shown by using Jacobsthal and Hadamard matrices. Some of these codes are also observed to be good equidistant code (Sinha et al., 2008).

The concatenated structure of quasi-abelian codes

Designs, Codes and Cryptography, 2021

The decomposition of a quasi-abelian code into shorter linear codes over larger alphabets was given in (Jitman, Ling, (2015)), extending the analogous Chinese remainder decomposition of quasi-cyclic codes (Ling, Solé, (2001)). We give a concatenated decomposition of quasi-abelian codes and show, as in the quasi-cyclic case, that the two decompositions are equivalent. The concatenated decomposition allows us to give a general minimum distance bound for quasi-abelian codes and to construct some optimal codes. Moreover, we show by examples that the minimum distance bound is sharp in some cases. In addition, examples of large strictly quasi-abelian codes of about a half rate are given. The concatenated structure also enables us to conclude that strictly quasi-abelian linear complementary dual codes over any finite field are asymptotically good.

Some constructions of linearly optimal group codes

Linear Algebra and its Applications, 2010

We continue here the research on (quasi)group codes over (quasi)group rings. We give some constructions of [n, n − 3, 3] qcodes over F q for n = 2q and n = 3q. These codes are linearly optimal, i.e. have maximal dimension among linear codes having a given length and distance. Although codes with such parameters are known, our main results state that we can construct such codes as (left) group codes. In the paper we use a construction of Reed-Solomon codes as ideals of the group ring F q G where G is an elementary abelian group of order q.

Perfect binary codes: constructions, properties, and enumeration

IEEE Transactions on Information Theory, 1994

Properties of nonlinear perfect binary codes are investigated and several new constructions of perfect codes are derived from these properties. An upper bound on the cardinality of the intersection of two perfect codes of length n is presented, and perfect codes whose intersection attains the upper bound are constructed for all n. As an immediate consequence of the proof of the upper bound we obtain a simple closed-form expression for the weight distribution of a perfect code. Furthermore, we prove that the characters of a perfect code satisfy certain constraints, and provide a sufficient condition for a binary code to be perfect. The latter result is employed to derive a generalization of the construction of Phelps, which is shown to give rise to some perfect codes that are nonequivalent to the perfect codes obtained from the known constructions. Moreover, for any m 2 4 we construct jidl-rank perfect binary codes of length 2" -1. These codes are obviously nonequivalent to any of the previously known perfect codes. Furthermore the latter construction exhibits the existence of full-rank perfect tilings. Finally, we construct a set of 22c" nonequivalent perfect codes of length n, for sufficiently large n and a constant c = 0.5 -E. Precise enumeration of the number of codes in this set provides a slight improvement over the results previously reported by Phelps.

Two theorems on perfect codes

Discrete Mathematics, 1972

Two theorems are proved on perfect Codes. The first one States that Lloyd's theorem is true without the assumption that tha number of Symbols m the alphabet is a pnme power The second theorem asseits the impossibiüty of perfect group codes over non-pnme-poweralphabets.

Computational Hardness and Explicit Constructions of Error Correcting Codes

Fourty-Fourth Annual …, 2006

We outline a procedure for using pseudorandom generators to construct binary codes with good properties, assuming the existence of sufficiently hard functions. Specifically, we give a polynomial time algorithm, which for every integers n and k, constructs polynomially many linear codes of block length n and dimension k, most of which achieving the Gilbert-Varshamov bound. The success of the procedure relies on the assumption that the exponential time class of E def = DTIME[2 O(n) ] is not contained in the sub-exponential space class DSPACE[2 o(n) ].

One–generator quasi–abelian codes revisited

The class of 1-generator quasi-abelian codes over finite fields is revisited. Alternative and explicit characterization and enumeration of such codes are given. An algorithm to find all 1-generator quasi-abelian codes is provided. Two 1-generator quasi-abelian codes whose minimum distances are improved from Grassl's online table are presented.

The rank and kernel of extended 1-perfect Z4-linear and additive non-Z4-linear codes

… Theory, IEEE Transactions on, 2003

A binary extended 1-perfect code of length n + 1 = 2 is additive if it is a subgroup of 2. The punctured code by deleting a coordinate (if there is one) gives a perfect additive code. 1-perfect additive codes were completely characterized in [3] and by using that characterization we compute the possible parameters , , rank, and dimension of the kernel for extended 1-perfect additive codes. A very special case is that of extended 1-perfect-linear codes.