A remark on full rank perfect codes

Any full rank perfect 1-error correcting binary code of length n=2^k-1 and with a kernel of dimension n-log(n+1)-m, where m is sufficiently large, may be used to construct a full rank perfect 1-error correcting binary code of length 2^m-1 and with a kernel of dimension n-log(n+1)-k. Especially we may construct full rank perfect 1-error correcting binary codes of length n=2^m-1 and with a kernel of dimension n-log(n+1)-4 for m=6,7,…,10.This result extends known results on the possibilities for the size of a kernel of a full rank perfect code.     

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.     

On the classification of perfect codes: side class structures

The side class structure of a perfect 1-error correcting binary code (hereafter referred to as a perfect code) C describes the linear relations between the coset representatives of the kernel of C. Two perfect codes C and C′ are linearly equivalent if there exists a non-singular matrix A such that AC = C′ where C and C′ are matrices with the code words of C and C′ as columns. Hessler proved that the perfect codes C and C′ are linearly equivalent if and only if they have isomorphic side class structures. The aim of this paper is to describe all side class structures. It is shown that the transpose of any side class structure is the dual of a subspace of the kernel of some perfect code and vice versa; any dual of a subspace of a kernel of some perfect code is the transpose of the side class structure of some perfect code. The conclusion is that for classification purposes of perfect codes it is sufficient to find the family of all kernels of perfect codes.     

A Full Rank Perfect Code of Length 31

A full rank perfect 1-error correcting binary code of length 31 with a kernel of dimension 21 is described. This was the last open case of the rank-kernel problem of Etzion and Vardy. AMS Classification: 94B25     

On the classification of perfect codes: Extended side class structures

The two 1-error correcting perfect binary codes, C and C^′ are said to be equivalent if there exists a permutation π of the set of the n coordinate positions and a word such that . Hessler defined C and C^′ to be linearly equivalent if there exists a non-singular linear map φ such that C^′=φ(C). Two perfect codes C and C^′ of length n will be defined to be extended equivalent if there exists a non-singular linear map φ and a word such that

     

A generalization of the Lee distance and error correcting codes

In this contribution, we will define an l-dimensional Lee distance which is a generalization of the Lee distance defined only over a prime field, and we will construct 2-error correcting codes for this distance. Our l-dimensional Lee distance can be defined not only over a prime field but also over any finite field. The ordinary Lee distance is just the one-dimensional Lee distance. Also the Mannheim or modular distances introduced by Huber are special cases of our distance.     

On rank and kernel of some mixed perfect codes

Mixed perfect 1-error correcting codes and the associated dual codes over the group Z(n,l),

     

Enumerating and decoding perfect linear Lee codes

Using group theory approach, we determine all numbers q for which there exists a linear 1-error correcting perfect Lee code of block length n over Z _q , and then we enumerate those codes. At the same time this approach allows us to design a linear time decoding algorithm.      

A coding problem in steganography

To study how to design a steganographic algorithm more efficiently, a new coding problem—steganographic codes (abbreviated stego-codes)—is presented in this paper. The stego-codes are defined over the field with q(q ≥ 2) elements. A method of constructing linear stego-codes is proposed by using the direct sum of vector subspaces. And the problem of linear stego-codes is converted to an algebraic problem by introducing the concept of the tth dimension of a vector space. Some bounds on the length of stego-codes are obtained, from which the maximum length embeddable (MLE) code arises. It is shown that there is a corresponding relation between MLE codes and perfect error-correcting codes. Furthermore the classification of all MLE codes and a lower bound on the number of binary MLE codes are obtained based on the corresponding results on perfect codes. Finally hiding redundancy is defined to value the performance of stego-codes.      

On perfect <Emphasis Type="Italic">p</Emphasis>-ary codes of length <Emphasis Type="Italic">p</Emphasis> + 1

Let p be a prime number and assume p ≥ 5. We will use a result of L. Redéi to prove, that every perfect 1-error correcting code C of length p + 1 over an alphabet of cardinality p, such that C has a rank equal to p and a kernel of dimension p − 2, will be equivalent to some Hamming code H. Further, C can be obtained from H, by the permutation of the symbols, in just one coordinate position.      

MDS and AMDS symbol-pair codes constructed from repeated-root cyclic codes

Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against the pair errors in symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. Based on repeated-root cyclic codes, we construct two classes of MDS symbol-pair codes for more general generator polynomials and also give a new class of almost MDS (AMDS) symbol-pair codes with the length lp. In addition, we derive all MDS and AMDS symbol-pair codes with length 3p, when the degree of the generator polynomials is no more than 10. The main results are obtained by determining the solutions of certain equations over finite fields.     

Construction of a (64, 2 37, 12) Code via Galois Rings

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4^m 16. At length 16 they coincide to give the Nordstrom-Robinson code. This paper constructs a nonlinear (64, 2₃₇, 12) code as the binary image, under the Gray map, of an extended cyclic code defined over the integers modulo 4 using Galois rings. The Nordstrom-Robinson code is defined in this same way, and like the Nordstrom-Robinson code, the new code is better than any linear code that is presently known.     

The Classification of Some Perfect Codes

Perfect 1-error correcting codes C in Z ₂ ⁿ , where n=2 ^m–1, are considered. Let ; denote the linear span of the words of C and let the rank of C be the dimension of the vector space . It is shown that if the rank of C is n–m+2 then C is equivalent to a code given by a construction of Phelps. These codes are, in case of rank n–m+2, described by a Hamming code H and a set of MDS-codes D _h , h H, over an alphabet with four symbols. The case of rank n–m+1 is much simpler: Any such code is a Vasil'ev code.     

Edge local complementation and equivalence of binary linear codes

Orbits of graphs under the operation edge local complementation (ELC) are defined. We show that the ELC orbit of a bipartite graph corresponds to the equivalence class of a binary linear code. The information sets and the minimum distance of a code can be derived from the corresponding ELC orbit. By extending earlier results on local complementation (LC) orbits, we classify the ELC orbits of all graphs on up to 12 vertices. We also give a new method for classifying binary linear codes, with running time comparable to the best known algorithm.     

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.     

On Perfect Codes: Rank and Kernel

We establish upper and lower bounds on the rank and the dimension of the kernel of perfect binary codes. We also establish some results on the structure of perfect codes.     

On perfect Lee codes

In this paper we survey recent results on the Golomb-Welch conjecture and its generalizations and variations. We also show that there are no perfect 2-error correcting Lee codes of block length 5 and 6 over Z. This provides additional support for the Golomb Welch conjecture as it settles the two smallest cases open so far.     

A Class of Perfect Ternary Constant-Weight Codes

We construct a class of perfect ternary constant-weight codes of length 2 ^r , weight 2 ^r -1 and minimum distance 3. The codes have codewords. The construction is based on combining cosets of binary Hamming codes. As a special case, for r=2 the construction gives the subcode of the tetracode consisting of its nonzero codewords. By shortening the perfect codes, we get further optimal codes.     

On linear complexity of binary lattices,II

The linear complexity is an important and frequently used measure of unpredictably and pseudorandomness of binary sequences. In Part I of this paper, we extended this notion to two dimensions: we defined and studied the linear complexity of binary and bit lattices. In this paper, first we will estimate the linear complexity of a truly random bit (M,N)-lattice. Next we will extend the notion of k-error linear complexity to bit lattices. Finally, we will present another alternative definition of linear complexity of bit lattices.     

Asymptotic analysis on the normalized <Emphasis Type="Italic">k</Emphasis>-error linear complexity of binary sequences

Linear complexity and k-error linear complexity are the important measures for sequences in stream ciphers. This paper discusses the asymptotic behavior of the normalized k-error linear complexity \({L_{n,k}(\underline{s})/n}\) of random binary sequences \({\underline{s}}\) , which is based on one of Niederreiter’s open problems. For k = n θ, where 0 ≤ θ ≤ 1/2 is a fixed ratio, the lower and upper bounds on accumulation points of \({L_{n,k}(\underline{s})/n}\) are derived, which holds with probability 1. On the other hand, for any fixed k it is shown that \({\lim_{n\rightarrow\infty} L_{n,k}(\underline{s})/n = 1/2}\) holds with probability 1. The asymptotic bounds on the expected value of normalized k-error linear complexity of binary sequences are also presented.