On a Class of Traceability Codes

Traceability codes are designed to be used in schemes that protect copyrighted digital data against piracy. The main aim of this paper is to give an answer to a Staddon–Stinson–Wei's problem of the existence of traceability codes with q< w ² and b>q. We provide a large class of these codes constructed by using a new general construction method for q-ary codes.     

MDS codes over finite principal ideal rings

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p ^e , l) (including ). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 ^e , l) of length n = 2 ^l for any a ≥ 1 and l ≥ 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.      

Reed-Muller codes and Barnes-Wall lattices: Generalized multilevel constructions and representation over GF(2<Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>)

Generalized multilevel constructions for binary RM(r,m) codes using projections onto GF(2 ^q ) are presented. These constructions exploit component codes over GF(2), GF(4),..., GF(2 ^q ) that are based on shorter Reed-Muller codes and set partitioning using partition chains of length-2 ^l codes. Using these constructions we derive multilevel constructions for the Barnes-Wall Λ(r,m) family of lattices which also use component codes over GF(2), GF(4),..., GF(2 ^q ) and set partitioning based on partition chains of length-2 ^l lattices. These constructions of Reed-Muller codes and Barnes-Wall lattices are readily applicable for their efficient decoding.      

New classes of self-complementary codes and quasi-symmetric designs

We present a new technique for constructing binary error correcting codes and give some examples of codes that can be constructed via this method. Among the examples is an infinite family of self-complementary codes with parameters (2u ²−u, 8u ², u ²−u) that can be constructed whenever there exists a u × u Hadamard Matrix. These codes meet the Grey–Rankin bound and imply the existence of quasi-symmetric designs on 2u ²−u points.     

Self-dual codes over the Kleinian four group

We introduce self-dual codes over the Kleinian four group K=Z ₂×Z ₂ for a natural quadratic form on K ⁿ and develop the theory. Topics studied are: weight enumerators, mass formulas, classification up to length 8, neighbourhood graphs, extremal codes, shadows, generalized t-designs, lexicographic codes, the Hexacode and its odd and shorter cousin, automorphism groups, marked codes. Kleinian codes form a new and natural fourth step in a series of analogies between binary codes, lattices and vertex operator algebras. This analogy will be emphasized and explained in detail.     

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.      

Cyclic Codes over the Integers Modulop

The purpose of this paper is twofold. First, we generalize the results of Pless and Qian and those of Pless, Solé, and Qian for cyclic ₄-codes to cyclic _p^m-codes. Second, we establish connections between this new development and the results on cyclic _p^m-codes obtained by Calderbank and Sloane. We produce generators for the cyclic _p^m-codes which are analogs to those for cyclic ₄-codes. We show that these may be used to produce a single generator for such codes. In particular, this proves that the ringR_n= _p^m[x]/(xⁿ− 1) is principal, a result that had been previously announced with an incorrect proof. Generators for dual codes of cyclic _p^m-codes are produced from the generators of the corresponding cyclic _p^m-codes. In addition, we also obtain generators for the cyclicp^m-ary codes induced from the idempotent generators for cyclicp-ary codes.     

Testing algebraic geometric codes

Property testing was initially studied from various motivations in 1990’s. A code C  GF (r)n is locally testable if there is a randomized algorithm which can distinguish with high possibility the codewords from a vector essentially far from the code by only accessing a very small (typically constant) number of the vector’s coordinates. The problem of testing codes was firstly studied by Blum, Luby and Rubinfeld and closely related to probabilistically checkable proofs (PCPs). How to characterize locally te...     

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.     

环F3+vF3上的循环码与常循环码

本文研究了环F₃+vF₃上的循环码与常循环码.通过环F₃+vF₃与域F₃上的循环码之间关系,证明了环F₃+vF₃上循环码是由一个多项式生成的.最后,用类似的方法,得到了环F₃+vF₃上v-常循环码也是由一个多项式生成的.     

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.      

Cyclic codes over the rings Z 2?+ uZ 2 and Z 2?+ uZ 2?+ u 2 Z 2

In this paper, we study cyclic codes over the rings Z ₂ + uZ ₂ and Z ₂ + uZ ₂ + u ² Z ₂ . We find a set of generators for these codes. The rank, the dual, and the Hamming distance of these codes are studied as well. Examples of cyclic codes of various lengths are also studied.      

New Extremal Type I Codes of Lengths 40, 42, and 44

It is known that it is possible to construct a generator matrix for a self-dual code of length 2n+2 from a generator matrix of a self-dual code of length 2n. With the aid of a computer, we construct new extremal Type I codes of lengths 40, 42, and 44 from extremal self-dual codes of lengths 38, 40, and 42 respectively. Among them are seven extremal Type I codes of length 44 whose weight enumerator is 1+224y ⁸+872y ¹⁰+·. A Type I code of length 44 with this weight enumerator was not known to exist previously.     

Cyclic codes over <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript>

Cyclic codes over an infinite family of rings are defined. The general properties of cyclic codes over these rings are studied, in particular nontrivial one-generator cyclic codes are characterized. It is also proved that the binary images of cyclic codes over these rings under the natural Gray map are binary quasi-cyclic codes of index 2 ^k . Further, several optimal or near optimal binary codes are obtained from cyclic codes over R _k via this map.     

On binary linear <Emphasis Type="Italic">r</Emphasis>-identifying codes

A subspace C of the binary Hamming space F ⁿ of length n is called a linear r-identifying code if for all vectors of F ⁿ the intersections of C and closed r-radius neighbourhoods are nonempty and different. In this paper, we give lower bounds for such linear codes. For radius r =  2, we give some general constructions. We give many (optimal) constructions which were found by a computer search. New constructions improve some previously known upper bounds for r-identifying codes in the case where linearity is not assumed.     

Artin–Schreier curves and weights of two-dimensional cyclic codes

Let be the finite field with q elements of characteristic p, be the extension of degree m>1 and f(x) be a polynomial over . The maximum number of affine -rational points that a curve of the form y^q−y=f(x) can have is q^m+1. We determine a necessary and sufficient condition for such a curve to achieve this maximum number. Then we study the weights of two-dimensional (2-D) cyclic codes. For this, we give a trace representation of the codes starting with the zeros of the dual 2-D cyclic code. This leads to a relation between the weights of codewords and a family of Artin–Schreier curves. We give a lower bound on the minimum distance for a large class of 2-D cyclic codes. Then we look at some special classes that are not covered by our main result and obtain similar minimum distance bounds.     

Negacyclic self-dual codes over finite chain rings

In this article, we study negacyclic self-dual codes of length n over a finite chain ring R when the characteristic p of the residue field [`(R)]{\bar{R}} and the length n are relatively prime. We give necessary and sufficient conditions for the existence of (nontrivial) negacyclic self-dual codes over a finite chain ring. As an application, we construct negacyclic MDR self-dual codes over GR(p ^t , m) of length p ^m  + 1.     

Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.      

Nets and their codes

A finitek-net of ordern is an incidence structure ofnk lines andn ² points, with any two lines either meeting once or being parallel, havingk parallel classes ofn lines each, and havingn points on each line. Finite nets are important to the study of finite planes and Latin squares.In this paper finite nets will be studied using the following linear codes: the row space of the incidence vectors of lines, the intersection of this code with its orthogonal, the code generated by differences of parallel lines, and the orthogonal to these codes. Using these codes we are able to recast the Moorhouse conjecture in terms of subcodes of the codes he uses, determine coding-theoretic reasons for a net's being maximal, and generalize a theorem of Assmus and Key which uses codes to classify finite planes of prime order.     

E-convex infix codes

Two partial orders that play an important role in the combinatorics of words, can be defined in a natural way on the free monoid X ^* generated by the finite alphabet X: the infix and the embedding orders. A set C of nonempty words is called an infix code (hypercode) over X if C is an antichain with respect to the infix (embedding) order. A set of words is said to be e-convex if it is convex with respect to the embedding order. Two characterizations of the e-convex infix codes are given as well as a sufficient condition for such codes to be finite. It is shown that the family EIC(X) of the e-convex infix codes with the empty word forms, under the operation of concatenation, a free submonoid of the free monoid B(X) of the biprefix codes and that the generating alphabet of EIC(X) is a sub-alphabet of the generating alphabet of B(X).This research was supported by Grant A7877 of the Natural Sciences and Engineering Council of Canada.