Self-dual codes over \mathbb{Z}_8 and \mathbb{Z}_9

We study self-dual codes over the rings and . We define various weights and weight enumerators over these rings and describe the groups of invariants for each weight enumerator over the rings. We examine the torsion codes over these rings to describe the structure of self-dual codes. Finally we classify self-dual codes of small lengths over .     

Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$

We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .      

The nonexistence of near-extremal formally self-dual codes

A code is called formally self-dual if and have the same weight enumerators. There are four types of nontrivial divisible formally self-dual codes over , and . These codes are called extremal if their minimum distances achieve the Mallows-Sloane bound. S. Zhang gave possible lengths for which extremal self-dual codes do not exist. In this paper, we define near-extremal formally self-dual (f.s.d.) codes. With Zhang’s systematic approach, we determine possible lengths for which the four types of near-extremal formally self-dual codes as well as the two types of near-extremal formally self-dual additive codes cannot exist. In particular, our result on the nonexistence of near-extremal binary f.s.d. even codes of any even length n completes all the cases since only the case 8|n was dealt with by Han and Lee.      

Maximal partial packings of $${Z_2^n}$$ with perfect codes

A maximal partial Hamming packing of is a family of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in . The number of translates of Hamming codes in is the packing number, and a partial Hamming packing is strictly partial if the family does not constitute a partition of . A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of , n = 2 ^m −1, satisfies . It is also proved that for any n equal to 2 ^m −1, , there exist maximal strictly partial Hamming packings of with packing numbers n−10,n−9,n−8,...,n−1. This implies that the upper bound is tight for any n = 2 ^m −1, . All packing numbers for maximal strictly partial Hamming packings of , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5,6,7,...,13 and 14.      

On circulant self-dual codes over small fields

We construct self-dual codes over small fields with q = 3, 4, 5, 7, 8, 9 of moderate length with long cycles in the automorphism group. With few exceptions, the codes achieve or improve the known lower bounds on the minimum distance of self-dual codes.      

On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.     

On self-dual codes over {\mathbb{F}}_5

The purpose of this paper is to improve the upper bounds of the minimum distances of self-dual codes over for lengths [22, 26, 28, 32–40]. In particular, we prove that there is no [22, 11, 9] self-dual code over , whose existence was left open in 1982. We also show that both the Hamming weight enumerator and the Lee weight enumerator of a putative [24, 12, 10] self-dual code over are unique. Using the building-up construction, we show that there are exactly nine inequivalent optimal self-dual [18, 9, 7] codes over up to the monomial equivalence, and construct one new optimal self-dual [20, 10, 8] code over and at least 40 new inequivalent optimal self-dual [22, 11, 8] codes.      

Self-dual bases in\mathbb{F}_{q^n }

We characterize weakly self-dual bases of the field extension over , examine the existence of weakly self-dual polynomial bases, and use duality to analyze normal basis multiplication.     

New Linear Codes Over \mathbb{F}_3 and \mathbb{F}_5 and Improvements on Bounds

One of the most important problems of coding theory is to constructcodes with best possible minimum distances. In this paper, we generalize the method introduced by [8] and obtain new codes which improve the best known minimum distance bounds of some linear codes. We have found a new linear ternary code and 8 new linear codes over with improved minimumdistances. First we introduce a generalized version of Gray map,then we give definition of quasi cyclic codes and introduce nearlyquasi cyclic codes. Next, we give the parameters of new codeswith their generator matrices. Finally, we have included twotables which give Hamming weight enumerators of these new codes.     

The number of self-dual codes over $${Z_{p^3}}$$

Let p be a prime number. In this paper, we consider codes over the ring of integers modulo p ³ and give a characterization of self-duality. This leads to a construction of self-dual codes and a mass formula, which counts the number of such codes over .      

\mathbb{Z }_2\mathbb{Z }_4-Additive formally self-dual codes

We study odd and even \(\mathbb{Z }_2\mathbb{Z }_4\) formally self-dual codes. The images of these codes are binary codes whose weight enumerators are that of a formally self-dual code but may not be linear. Three constructions are given for formally self-dual codes and existence theorems are given for codes of each type defined in the paper.     

Extremal Self-Dual Codes over \mathbb{F} 2 × \mathbb{F} 2

In this paper, it is shown that extremal (Hermitian) self-dual codes over ₂ × ₂ exist only for lengths 1, 2, 3, 4, 5, 8 and 10. All extremal self-dual codes over ₂ × ₂ are found. In particular, it is shown that there is a unique extremal self-dual code up to equivalence for lengths 8 and 10. Optimal self-dual codes are also investigated. A classification is given for binary [12, 7, 4] codes with dual distance 4, binary [13, 7, 4] codes with dual distance 4 and binary [13, 8, 4] codes with dual distance 4.     

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps G to the standard basis of . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space , which does not assume an “a priori” group algebra structure on . As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed–Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes. Research supported by D.G.I. of Spain and Fundación Séneca of Murcia.     

Codes over \mathbb{F}_3 + u\mathbb{F}_3 and Improvements to the Bounds on Ternary Linear Codes

New ternary linear codeswith parameters [208, 8, 127], [150, 10, 85],[160, 10, 91], [170, 10, 97], [180,10, 103], and [190, 10, 110], are found whichimprove the known lower bound on the maximum possible minimumHamming distance. These codes are constructed from codes over via a Gray map.     

There exists no self-dual [24,12,10] code over $${{\mathbb F}_5}$$

Self-dual codes over exist for all even lengths. The smallest length for which the largest minimum weight among self-dual codes has not been determined is 24, and the largest minimum weight is either 9 or 10. In this note, we show that there exists no self-dual [24, 12, 10] code over , using the classification of 24-dimensional odd unimodular lattices due to Borcherds.      

Type II Self-Dual Codes over Finite Rings and Even Unimodular Lattices

In this paper, we investigate self-dual codes over finite rings, specifically the ring of integers modulo 2m. Type II codes over are introduced as self-dual codes with Euclidean weights which are a multiple of 2m +1. We describe a relationship between Type II codes and even unimodular lattices. This relationship provides much information on Type II codes. Double circulant Type II codes over are also studied.     

Cyclic DNA codes over the ring $$\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2+v^2\mathbb {F}_2+uv^2\mathbb {F}_2$$

We study the structure of cyclic DNA codes of odd length over the finite commutative ring \(R=\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2 + v^2\mathbb {F}_2+uv^2\mathbb {F}_2,~u^2=0, v^3=v\), which plays an important role in genetics, bioengineering and DNA computing. A direct link between the elements of the ring R and 64 codons used in the amino acids of living organisms is established by introducing a Gray map from R to \(R_1=\mathbb {F}_2+u\mathbb {F}_2 ~(u^2=0)\). The reversible and the reversible-complement codes over R are investigated. We also discuss the binary image of the cyclic DNA codes over R. Among others, some examples of DNA codes obtained via Gray map are provided.     

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

The Automorphism Groups of BCH Codes and of Some Affine-Invariant Codes Over Extension Fields

Affine-invariant codes are extended cyclic codes of length p ^m invariant under the affine-group acting on . This class of codes includes codes of great interest such as extended narrow-sense BCH codes. In recent papers, we classified the automorphism groups of affine-invariant codes berg, bech1. We derive here new results, especially when the alphabet field is an extension field, by expanding our previous tools. In particular we complete our results on BCH codes, giving the automorphism groups of extended narrow-sense BCH codes defined over any extension field.     

Double Circulant Codes over \mathbb{Z}_4 and Even Unimodular Lattices

With the help of some new results about weight enumerators of self-dual codes over we investigate a class of double circulant codes over , one of which leads to an extremal even unimodular 40–dimensional lattice. It is conjectured that there should be Nine more constructions of the Leech lattice