A generalized Gleason–Pierce–Ward theorem

The Gleason–Pierce–Ward theorem gives constraints on the divisor and field size of a linear divisible code over a finite field whose dimension is half of the code length. This result is a departure point for the study of self-dual codes. In recent years, additive codes have been studied intensively because of their use in additive quantum codes. In this work, we generalize the Gleason–Pierce–Ward theorem on linear codes over GF(q), q = p ^m , to additive codes over GF(q). The first step of our proof is an application of a generalized upper bound on the dimension of a divisible code determined by its weight spectrum. The bound is proved by Ward for linear codes over GF(q), and is generalized by Liu to any code as long as the MacWilliams identities are satisfied. The trace map and an analogous homomorphism on GF(q) are used to complete our proof.      

On Intersection Problem for Perfect Binary Codes

The main result is that to any even integer q in the interval 0 ≤ q ≤ 2^{n+1-2log(n+1)}, there are two perfect codes C₁ and C₂ of length n = 2^m − 1, m ≥ 4, such that |C₁ ∩ C₂| = q.     

Direct constructions of additive codes

A code is q^m‐ary q‐linear if its alphabet forms an m‐dimensional vector space over ??_q and the code is linear over ??_q. These additive codes form a natural generalization of linear codes. Our main results are direct constructions of certain families of additive codes. These comprise the additive generalization of the Kasami codes, an additive generalization of the Bose‐Bush construction of orthogonal arrays of strength 2 as well as a class of additive codes which are being used for deep space communication. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 207–216, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.20000     

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.     

On the maximality of linear codes

We show that if a linear code admits an extension, then it necessarily admits a linear extension. There are many linear codes that are known to admit no linear extensions. Our result implies that these codes are in fact maximal. We are able to characterize maximal linear (n, k, d) _q -codes as complete (weighted) (n, n − d)-arcs in PG(k − 1, q). At the same time our results sharply limit the possibilities for constructing long non-linear codes. The central ideas to our approach are the Bruen-Silverman model of linear codes, and some well known results on the theory of directions determined by affine point-sets in PG(k, q).      

Constructions and bounds on linear error-block codes

We obtain new bounds on the parameters and we give new constructions of linear error-block codes. We obtain a Gilbert–Varshamov type construction. Using our bounds and constructions we obtain some infinite families of optimal linear error-block codes over . We also study the asymptotic of linear error-block codes. We define the real valued function α _q,m,a (δ), which is an analog of the important real valued function α _q (δ) in the asymptotic theory of classical linear error-correcting codes. We obtain both Gilbert–Varshamov and algebraic geometry type lower bounds on α _q,m,a (δ). We compare these lower bounds in graphs.      

Negacyclic duadic codes

This paper extends the concepts from cyclic duadic codes to negacyclic codes over F_q (q an odd prime power) of oddly even length. Generalizations of defining sets, multipliers, splittings, even-like and odd-like codes are given. Necessary and sufficient conditions are given for the existence of self-dual negacyclic codes over F_q and the existence of splittings of 2N, where N is odd. Other negacyclic codes can be extended by two coordinates in a way to create self-dual codes with familiar parameters.     

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.     

Kernels and <Emphasis Type="Italic">p</Emphasis>-Kernels of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>-ary 1-Perfect Codes

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over as well as over the prime field , are established. Q-ary 1-perfect codes of length n=(q^m − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.Communicated by: I.F. Blake     

Pseudogeometries with clusters and an example of a recursive [4, 2, 3]<Subscript>42</Subscript>-code

In 1998, E. Couselo, S. Gonzalez, V. Markov, and A. Nechaev defined the recursive codes and obtained some results that allowed one to conjecture the existence of recursive MDS-codes of dimension 2 and length 4 over any finite alphabet of cardinality q ∉ {2, 6}. This conjecture remained open only for q ∈ {14, 18, 26, 42}. It is shown in this paper that there exist such codes for q = 42. We used a new construction, that of pseudogeometry with clusters.     

Enumeration of inequivalent irreducible Goppa codes

We consider irreducible Goppa codes over F_q of length qⁿ defined by polynomials of degree r, where q is a prime power and n,r are arbitrary positive integers. We obtain an upper bound on the number of such codes.     

An Extension Theorem for Linear Codes

One of the first results one meets in coding theory is that a binary linear [n,k,d] code, whose minimum distance is odd, can be extended to an [n + 1, k, d + 1] code. This is one of the few elementary results about binary codes which does not obviously generalise to q-ary codes. The aim of this paper is to give a simple sufficient condition for a q-ary [n, k, d] code to be extendable to an [n + 1, k, d + 1] code. Applications will be given to the construction and classification of good codes, to proving the non- existence of certain codes, and also an application in finite geometry.     

Small weight codewords in the codes arising from Desarguesian projective planes

We study codewords of small weight in the codes arising from Desarguesian projective planes. We first of all improve the results of K. Chouinard on codewords of small weight in the codes arising from PG(2, p), p prime. Chouinard characterized all the codewords up to weight 2p in these codes. Using a particular basis for this code, described by Moorhouse, we characterize all the codewords of weight up to 2p + (p−1)/2 if p ≥ 11. We then study the codes arising from . In particular, for q ₀ = p prime, p ≥ 7, we prove that the codes have no codewords with weight in the interval [q + 2, 2q − 1]. Finally, for the codes of PG(2, q), q = p ^h , p prime, h ≥ 4, we present a discrete spectrum for the weights of codewords with weights in the interval [q + 2, 2q − 1]. In particular, we exclude all weights in the interval [3q/2, 2q − 1]. Geertrui Van de Voorde research is supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen) Joost Winne was supported by the Fund for Scientific Research - Flanders (Belgium).     

Constructions of partitions of FqN into perfect q-ary codes

Given N = (q ^m − 1)/(q − 1), where q is a power of a prime, q > 2, we present two constructions of different partitions of the set F _q ^N of all q-ary length N vectors into perfect q-ary codes of length N. The lower bounds on the number of these partitions are presented.     

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.     

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.      

Symplectic spread-based generalized Kerdock codes

Kerdock codes (Kerdock, Inform Control 20:182–187, 1972) are a well-known family of non-linear binary codes with good parameters admitting a linear presentation in terms of codes over the ring (see Nechaev, Diskret Mat 1:123–139, 1989; Hammons et al., IEEE Trans Inform Theory 40:301–319, 1994). These codes have been generalized in different directions: in Calderbank et al. (Proc Lond Math Soc 75:436–480, 1997) a symplectic construction of non-linear binary codes with the same parameters of the Kerdock codes has been given. Such codes are not necessarily equivalent. On the other hand, in Kuzmin and Nechaev (Russ Math Surv 49(5), 1994) the authors give a family of non-linear codes over the finite field F of q = 2 ^l elements, all of them admitting a linear presentation over the Galois Ring R of cardinality q ² and characteristic 2². The aim of this article is to merge both approaches, obtaining in this way new families of non-linear codes over F that can be presented as linear codes over the Galois Ring R. The construction uses symplectic spreads.      

The Moment Problem Associated with the q -Laguerre Polynomials

   Abstract. We consider the indeterminate Stieltjes moment problem associated with the q -Laguerre polynomials. A transformation of the set of solutions, which has all the classical solutions as fixed points, is established and we present a method to construct, for instance, continuous singular solutions. The connection with the moment problem associated with the Stieltjes—Wigert polynomials is studied; we show how to come from q -Laguerre solutions to Stieltjes—Wigert solutions by letting the parameter α —> ∞ , and we explain how to lift a Stieltjes—Wigert solution to a q -Laguerre solution at the level of Pick functions. Based on two generating functions, expressions for the four entire functions from the Nevanlinna parametrization are obtained.     

The nonexistence of ternary [105,6,68] and [230,6,152] codes

Let [n,k,d]_q-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). In this paper, the nonexistence of [105,6,68]₃ and [230,6,152]₃ codes is proved.