Perfect codes as isomorphic spaces

Linear equivalence between perfect codes is defined. This definition gives the concept of general perfect 1-error correcting binary codes. These are defined as 1-error correcting perfect binary codes, with the difference that the set of errors is not the set of weight one words, instead any set with cardinality n and full rank is allowed. The side class structure defines the restrictions on the subspace of any general 1-error correcting perfect binary code. Every linear equivalence class will contain all codes with the same length, rank and dimension of kernel and all codes in the linear equivalence class will have isomorphic side class structures.     

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.     

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     

Switching Equivalence Classes of Perfect Codes

The authors present a 1-error correcting perfect code of length 15 and show that it is not switching equivalent to the Hamming code thereby settling a question of Avgustinovich and Solov'evaas96     

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.      

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.     

Embedding in a perfect code

A binary 1 ‐error‐correcting code can always be embedded in a 1 ‐perfect code of some larger length. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 419–423, 2009     

Weighted Hamming metric structures

It is known that the binary generalized Goppa codes are perfect codes for the weighted Hamming metrics. In this paper, we present the existence of a weighted Hamming metric that admits a binary Hamming code (resp. an extended binary Hamming code) to be perfect code. For a special weighted Hamming metric, we also give some structures of a 2-perfect code, show how to construct a 2-perfect linear code and obtain the weight distribution of a 2-perfect code from the partial information of the code.     

On the construction of perfect deletion-correcting codes using design theory

We consider a way to construct perfect codes capable of correcting 2 or more deletions using design-theory. As a starting point we use an (ordered) block design to construct a perfect deletion correcting code. Using this code we are able to construct more perfect deletion correcting codes over smaller or larger alphabets by removing or adding symbols in a smart way.In this way we are able to find all perfect 2-deletion correcting codes of length 4, and all perfect 3-deletion correcting codes of length 5 with different coordinates. The perfect 3-deletion correcting codes of length 5 with repeated symbols can be constructed for almost all possible alphabet sizesv, except forv=13, 14, 15, and 16, and forv7, 8 (mod 10),v17. For these values ofv we are neither able to prove the existence, nor the non-existence of perfect 3-deletion correcting codes of length 5 over an alphabet of sizev.     

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.      

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 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.     

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),

     

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.     

On perfect deletion‐correcting codes

In this article, we present constructions for perfect deletion‐correcting codes. The first construction uses perfect deletion‐correcting codes without repetition of letters to construct other perfect deletion‐correcting codes. This is a generalization of the construction shown in 1 . In the third section, we investigate several constructions of perfect deletion‐correcting codes using designs. In the last section, we investigate perfect deletion‐correcting codes containing few codewords. © 2003 Wiley Periodicals, Inc.     

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     

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 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.     

An improved version of Lloyd's theorem

The generalisation of Lloyd's theorem to distance-transitive graphs can be improved in the case of antipodal graphs by looking at the derived graph. In the case of binary perfect codes the roots of the Lloyd polynomial are even integers. This can be applied to give a short proof of the binary perfect code theorem.