On non-antipodal binary completely regular codes

Binary non-antipodal completely regular codes are characterized. Using a result on nonexistence of nontrivial binary perfect codes, it is concluded that there are no unknown nontrivial non-antipodal completely regular binary codes with minimum distance d?3. The only such codes are halves and punctured halves of known binary perfect codes. Thus, new such codes with covering radius ρ=6 and 7 are obtained. In particular, a half of the binary Golay [23,12,7]-code is a new binary completely regular code with minimum distance d=8 and covering radius ρ=7. The punctured half of the Golay code is a new completely regular code with minimum distance d=7 and covering radius ρ=6. The new code with d=8 disproves the known conjecture of Neumaier, that the extended binary Golay [24,12,8]-code is the only binary completely regular code with d?8. Halves of binary perfect codes with Hamming parameters also provide an infinite family of binary completely regular codes with d=4 and ρ=3. Puncturing of these codes also provide an infinite family of binary completely regular codes with d=3 and ρ=2. Both these families of codes are well known, since they are uniformly packed in the narrow sense, or extended such codes. Some of these completely regular codes are new completely transitive codes.     

Perfect binary codes of infinite length

A subset C of infinite-dimensional binary cube is called a perfect binary code with distance 3 if all balls of radius 1 (in the Hamming metric) with centers in C are pairwise disjoint and their union cover this binary cube. Similarly, we can define a perfect binary code in zero layer, consisting of all vectors of infinite-dimensional binary cube having finite supports. In this article we prove that the cardinality of all cosets of perfect binary codes in zero layer is the cardinality of the continuum. Moreover, the cardinality of all cosets of perfect binary codes in the whole binary cube is equal to the cardinality of the hypercontinuum.     

Class of generalized Goppa codes perfect in weighted Hamming metric

A weighted Hamming metric is considered. A class of binary linear codes consistent with the weighted Hamming metric is discussed. A class of binary generalized Goppa codes perfect in the weighted Hamming metric is offered.     

On kernels of perfect codes

It is shown that there exists a perfect one-error-correcting binary code with a kernel which is not contained in any Hamming code.     

On full-rank perfect codes over finite fields

We propose a construction of full-rank q-ary 1-perfect codes. This is a generalization of the construction of full-rank binary 1-perfect codes by Etzion and Vardy (1994). The properties of the i-components of q-ary Hamming codes are investigated, and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1-perfect codes is generalized to the q-ary case. We propose a generalization of the notion of an i-component of a 1-perfect code and introduce the concept of an (i, σ)-component of a q-ary 1-perfect code. We also present a generalization of the Lindström–Schönheim construction of q-ary 1-perfect codes and provide a lower bound for the number of pairwise distinct q-ary 1-perfect codes of length n.     

On the binary codes with parameters of triply-shortened 1-perfect codes

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary (n?=?2 ^m ? 3, 2 ^n-m-1, 4) code C, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the n-cube into six cells. An arbitrary binary (n?=?2 ^m ? 4, 2 ^n-m , 3) code D, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) with six cells. As a corollary, the codes C and D are completely semiregular; i.e., the weight distribution of such codes depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if D is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable.     

New bounds on binary identifying codes

The original motivation for identifying codes comes from fault diagnosis in multiprocessor systems. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks.In this paper, we concentrate on identification in binary Hamming spaces. We give a new lower bound on the cardinality of r-identifying codes when r≥2. Moreover, by a computational method, we show that M₁(6)=19. It is also shown, using a non-constructive approach, that there exist asymptotically good (r,≤?)-identifying codes for fixed ?≥2. In order to construct (r,≤?)-identifying codes, we prove that a direct sum of r codes that are (1,≤?)-identifying is an (r,≤?)-identifying code for ?≥2.     

Non-invertible-element constacyclic codes over finite PIRs

In this paper we introduce the notion of λ-constacyclic codes over finite rings R for arbitrary element λ of R. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings, give the unique form of the sets of the defining polynomials and obtain their minimum Hamming distances. A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided. In particular, we give a necessary and sufficient condition for the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the Chinese Remainder Theorem, we study the NIE-constacyclic codes over finite PIRs. Furthermore, we construct some optimal NIE-constacyclic codes over finite PIRs in the sense that they achieve the maximum possible minimum Hamming distances for some given lengths and cardinalities.     

Weighted perfect codes in Lee metric

In this talk, we will present results about perfect weighted coverings of radius 1 in the Lee metric. Weighted coverings are a very natural generalization of many classes of codes. Perfect weighted coverings are well studied in the Hamming metric, but also in other contexts with different names, such as regular-sets, multiple coverings or [a, b]-dominating sets. In this talk, we present results of existence as well as of non-existence for perfect weighted coverings of radius one on the multidimensional grid graphs.     

On the admissible families of components of hamming codes

In this paper, the properties of the i-components of Hamming codes are described. We suggest constructions of the admissible families of components of Hamming codes. Each q-ary code of length m and minimum distance 5 (for q = 3, the minimum distance is 3) is shown to embed in a q-ary 1-perfect code of length n = (q ^m − 1)/(q − 1). Moreover, each binary code of length m+k and minimum distance 3k + 3 embeds in a binary 1-perfect code of length n = 2 ^m − 1.     

On the Steiner quadruple systems of small rank which are embeddable into extended perfect binary codes

The codewords of weight 4 of every extended perfect binary code that contains the all-zero vector are known to form a Steiner quadruple system. We propose a modification of the Lindner construction for the Steiner quadruple system of order N = 2 ^r which can be described by special switchings from the Hamming Steiner quadruple system. We prove that each of these Steiner quadruple systems is embedded into some extended perfect binary code constructed by the method of switching of ijkl-components from the binary extended Hamming code. We give the lower bound for the number of different Steiner quadruple systems of order N with rank at most N ? logN + 1 which are embedded into extended perfect codes of length N.     

The poset structures admitting the extended binary Golay code to be a perfect code

Brualdi et al. [Codes with a poset metric, Discrete Math. 147 (1995) 57-72] introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Golay code to be a 4-error-correcting perfect P-code. In this paper we classify all of the poset structures which admit the extended binary Golay code to be a 4-error-correcting perfect P-code, and show that there are no posets which admit the extended binary Golay code to be a 5-error-correcting perfect P-code.     

On enumeration of nonequivalent perfect binary codes of length 15 and rank 15

All nonequivalent perfect binary codes of length 15 and rank 15 are constructed that are obtained from the Hamming code H ¹⁵ by translating its disjoint components. Also, the main invariants of this class of codes are determined such as the ranks, dimensions of kernels, and orders of automorphism groups.     

卡氏积码的r-MDR码与P_r-MDS码

本文研究了卡氏积码的r-广义Hamming重量计算公式和广义Singleton界,利用r-卡氏积码的子码仍为卡氏积码,证明了r-MDR码或Pr-MDR码的卡氏积码仍为r-MDR码或Pr-MDR码.同时也给出了这一个结果的部分逆命题.     

Optimal binary covering codes of length 2j

The minimum size of a binary covering code of length n and covering radius r is denoted by K(n,r), and codes of this length are called optimal. For j > 0 and n = 2^j, it is known that K(n,1) = 2 · K(n?1,1) = 2^{n ? j}. Say that two binary words of length n form a duo if the Hamming distance between them is 1 or 2. In this paper, it is shown that each optimal binary covering code of length n = 2^j, j > 0, and covering radius 1 is the union of duos in just one way, and that the closed neighborhoods of the duos form a tiling of the set of binary words of length n. Methods of constructing such optimal codes from optimal covering codes of length n ? 1 (that is, perfect single‐error‐correcting codes) are discussed. The paper ends with the construction of an optimal covering code of length 16 that does not contain an extension of any optimal covering code of length 15. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Weight distributions and weight hierarchies of two classes of binary linear codes

Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of a linear code has many applications such as on the type II wire-tap channel, dealing with t-resilient functions and trellis or branch complexity of linear codes and so on. In this paper, we present a formula for computing the weight hierarchies of linear codes constructed by the generalized method of defining sets. Then, we construct two classes of binary linear codes with a few weights and determine their weight distributions and weight hierarchies completely. Some codes of them can be used in secret sharing schemes.     

Perfect codes from the dual point of view I

The two concepts dual code and parity check matrix for a linear perfect 1-error correcting binary code are generalized to the case of non-linear perfect codes. We show how this generalization can be used to enumerate some particular classes of perfect 1-error correcting binary codes. We also use it to give an answer to a problem of Avgustinovich: whether or not the kernel of every perfect 1-error correcting binary code is always contained in some Hamming code.     

Trellis Structure and Higher Weights of Extremal Self-Dual Codes

A method for demonstrating and enumerating uniformly efficient (permutation-optimal) trellis decoders for self-dual codes of high minimum distance is developed. Such decoders and corresponding permutations are known for relatively few codes.The task of finding such permutations is shown to be substantially simplifiable in the case of self-dual codes in general, and for self-dual codes of sufficiently high minimum distance it is shown that it is frequently possible to deduce the existence of these permutations directly from the parameters of the code.A new and tighter link between generalized Hamming weights and trellis representations is demonstrated: for some self-dual codes, knowledge of one of the generalized Hamming weights is sufficient to determine the entire optimal state complexity profile.These results are used to characterize the permutation-optimal trellises and generalized Hamming weights for all [32,16,8] binary self-dual codes and for several other codes. The numbers of uniformly efficient permutations for several codes, including the [24,12,8] Golay code and both [24,12,9] ternary self-dual codes, are found.