On the minimum average distance of binary constant weight codes

Let β(n,M,w) denote the minimum average Hamming distance of a binary constant weight code with length n, size M and weight w. In this paper, we study the problem of determining β(n,M,w). Using the methods from coding theory and linear programming, we derive several lower bounds on the average Hamming distance of a binary constant weight code. These lower bounds enable us to determine the exact value for β(n,M,w) in several cases.     

Two new bounds on the size of binary codes with a minimum distance of three

We prove two new upper bounds on the size of binary codes with a minimum distance of three, namelyA(10, 3)76 andA(11, 3)152.     

Johnson type bounds on constant dimension codes

Very recently, an operator channel was defined by Koetter and Kschischang when they studied random network coding. They also introduced constant dimension codes and demonstrated that these codes can be employed to correct errors and/or erasures over the operator channel. Constant dimension codes are equivalent to the so-called linear authentication codes introduced by Wang, Xing and Safavi-Naini when constructing distributed authentication systems in 2003. In this paper, we study constant dimension codes. It is shown that Steiner structures are optimal constant dimension codes achieving the Wang-Xing-Safavi-Naini bound. Furthermore, we show that constant dimension codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures. Then, we derive two Johnson type upper bounds, say I and II, on constant dimension codes. The Johnson type bound II slightly improves on the Wang-Xing-Safavi-Naini bound. Finally, we point out that a family of known Steiner structures is actually a family of optimal constant dimension codes achieving both the Johnson type bounds I and II.      

Lower bounds on the minimum distance in Hermitian one-point differential codes

Korchmáros and Nagy [Hermitian codes from higher degree places. J Pure Appl Algebra, doi: 10. 1016/j.jpaa.2013.04.002, 2013] computed the Weierstrass gap sequence G(P) of the Hermitian function field Fq2( H ) at any place P of degree 3, and obtained an explicit formula of the Matthews-Michel lower bound on the minimum distance in the associated differential Hermitian code CΩ(D, mP ) where the divisor D is, as usual, the sum of all but one 1-degree Fq2-rational places of Fq2( H ) and m is a positive integer. For plenty of values of m depending on q, this provided improvements on the designed minimum distance of CΩ(D, mP). Further improvements from G(P) were obtained by Korchmáros and Nagy relying on algebraic geometry. Here slightly weaker improvements are obtained from G(P) with the usual function-field method depending on linear series, Riemann-Roch theorem and Weierstrass semigroups. We also survey the known results on this subject.     

Improved Delsarte bounds for spherical codes in small dimensions

We present an extension of the Delsarte linear programming method for spherical codes. For several dimensions it yields improved upper bounds including some new bounds on kissing numbers. Musin's recent work on kissing numbers in dimensions three and four can be formulated in our framework.     

Improving bounds on the minimum Euclidean distance for block codes by inner distance measure optimization

The minimum Euclidean distance is a fundamental quantity for block coded phase shift keying (PSK). In this paper we improve the bounds for this quantity that are explicit functions of the alphabet size q, block length n and code size |C|. For q=8, we improve previous results by introducing a general inner distance measure allowing different shapes of a neighborhood for a codeword. By optimizing the parameters of this inner distance measure, we find sharper bounds for the outer distance measure, which is Euclidean.The proof is built upon the Elias critical sphere argument, which localizes the optimization problem to one neighborhood. We remark that any code with q=8 that fulfills the bound with equality is best possible in terms of the minimum Euclidean distance, for given parameters n and |C|. This is true for many multilevel codes.     

Lower bounds on covering codes via partition matrices

Let Kq(n,R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Let σq(n,s;r) denote the minimal cardinality of a q-ary code of length n, which is s-surjective with radius r. In order to lower-bound Kq(n,n−2) and σq(n,s;s−2) we introduce partition matrices and their transversals. Our approach leads to a short new proof of a classical bound of Rodemich on Kq(n,n−2) and to the new bound Kq(n,n−2)?3q−2n+2, improving the first iff 5?n<q?2n−4. We determine Kq(q,q−2)=q−2+σ₂(q,2;0) if q?10. Moreover, we obtain the new powerful recursive bound Kq+1(n+1,R+1)?min{2(q+1),Kq(n,R)+1}.     

On distance nonregularity of Preparata codes

It is shown that among all Preparata codes only the code of length 16 is distance regular. An analogous result takes place for Preparata codes after puncturing any coordinate (only the code of length 15 is distance regular).     

AMDS symbol-pair codes from repeated-root cyclic codes

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance is of significance in determining the error-correcting capability of a symbol-pair code. One of the central themes in symbol-pair coding theory is the constructions of symbol-pair codes with the largest possible minimum symbol-pair distance. Maximum distance separable (MDS) and almost maximum distance separable (AMDS) symbol-pair codes are optimal and sub-optimal regarding the Singleton bound, respectively. In this paper, six new classes of AMDS symbol-pair codes are explicitly constructed through repeated-root cyclic codes. Remarkably, one class of such codes has unbounded lengths and the minimum symbol-pair distance of another class can reach 13.     

Schubert varieties, linear codes and enumerative combinatorics

We consider linear error correcting codes associated to higher-dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have recently been studied. The basic parameters such as the length, dimension and minimum distance of these codes are known only in special cases. An upper bound for the minimum distance is known and it is conjectured that this bound is achieved. We give explicit formulae for the length and dimension of arbitrary Schubert codes and prove the minimum distance conjecture in the affirmative for codes associated to Schubert divisors.     

Sharp bounds on distance spectral radius of graphs

Let D(G) denote the distance matrix of a connected graph G. The largest eigenvalue of D(G) is called the distance spectral radius of a graph G, denoted by ?(G). In this article, we give sharp upper and lower bounds for the distance spectral radius and characterize those graphs for which these bounds are best possible.     

The asymptotic probability distribution of the relative distance of additive quantum codes

For given information rate R, it is proved as n tends to infinite, that almost all additive ?n,n⋅R? quantum codes (pure and impure) are the codes with their relative distance tending to h⁻¹(1−R), where is an entropy function.     

Improved bounds for interatomic distance in Morse clusters

We improve the best known lower bounds on the distance between two points of an optimal Morse cluster, with ρ[4.967,15]. We develop a generalization of a method previously applied to the Lennard-Jones potential, that also leads to improvements of lower bounds for the Morse potential.     

The characterization of binary constant weight codes meeting the bound of Fu and Shen

Fu and Shen gave an upper bound on binary constant weight codes. In this paper, we present a new proof for the bound of Fu and Shen and characterize binary constant weight codes meeting this bound. It is shown that binary constant weight codes meet the bound of Fu and Shen if and only if they are generated from certain symmetric designs and quasi-symmetric designs in combinatorial design theory. In particular, it turns out that the existence of binary codes with even length meeting the Grey–Rankin bound is equivalent to the existence of certain binary constant weight codes meeting the bound of Fu and Shen. Furthermore, some examples are listed to illustrate these results. Finally, we obtain a new upper bound on binary constant weight codes which improves on the bound of Fu and Shen in certain case. This research is supported in part by the DSTA research grant R-394-000-025-422 and the National Natural Science Foundation of China under the Grant 60402031, and the NSFC-GDSF joint fund under the Grant U0675001     

Fuzzy codes and distance properties

A fuzzy code is defined as a fuzzy subset of n-tuples over a set F. An analysis of the Hamming distance between two fuzzy codewords and the error-correcting capability of a regular code in terms of its corresponding fuzzy code is presented.     

Several classes of optimal ternary cyclic codes with minimal distance four

In this paper, by analyzing the solutions of certain equations over ${\mathbb{F}}_{3^m}$, we present four classes of optimal ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$. It is shown that some recent work on this class of optimal ternary cyclic codes are special cases of our results.     

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.