On the Construction of Multiply Constant-Weight Codes

Multiply constant-weight codes (MCWCs) were introduced recently to improve the reliability of certain physically unclonable function response. In this paper, two methods of constructing MCWCs are presented following the concatenation methodology. In other words, MCWCs are constructed by concatenating approximate outer codes and inner codes. Besides, several classes of optimal MCWCs are derived from these methods. In the first method, the outer codes are [Formula: see text]-ary codes and the inner codes are constant-weight codes over [Formula: see text]. Furthermore, if the outer code achieves the Plotkin bound and the inner code achieves Johnson bound, then the resulting MCWC is optimal. In the second method, the outer codes are [Formula: see text]-ary codes and the inner codes are MCWCs. Furthermore, if the outer code achieves the Plotkin bound and the inner code achieves the Johnson bound, then the resulting MCWC is optimal.

Johnson Type Bounds for Mixed Dimension Subspace Codes

Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.

Bounds on Subspace Codes Based on Orthogonal Space Over Finite Fields of Characteristic 2

In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length [Formula: see text], size [Formula: see text], minimum subspace distance [Formula: see text] based on [Formula: see text]-dimensional totally singular subspace in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over finite fields [Formula: see text] of characteristic 2, denoted by [Formula: see text], are presented, where [Formula: see text] is a positive integer, [Formula: see text], [Formula: see text], [Formula: see text]. Then, we prove that [Formula: see text] codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in [Formula: see text], where [Formula: see text] denotes the collection of all the [Formula: see text]-dimensional totally singular subspaces in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over [Formula: see text] of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code [Formula: see text] in [Formula: see text] are provided, where [Formula: see text] denotes the collection of all the totally singular subspaces in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over [Formula: see text] of characteristic 2.

Some sequences of optimal constant weight codes

We introduce some sequences of binary constant weight codes which are constructed from the affine constructions of balanced incomplete block designs. They give the values for the function [Formula: see text] of constant weight codes, and they are optimal since those values reach the Johnson Bound. We focus on the values obtained by this method.

Bounds on subspace codes based on totally isotropic subspaces in unitary spaces

In this paper, the Sphere-packing bound, Singleton bound, Wang–Xing–Safavi-Naini bound, Johnson bound and Gilbert–Varshamov bound on the subspace codes [Formula: see text] based on [Formula: see text]-dimensional totally isotropic subspaces in unitary space [Formula: see text] over finite fields [Formula: see text] are presented. Then, we prove that [Formula: see text] codes based on [Formula: see text]-dimensional totally isotropic subspaces in unitary space [Formula: see text] attain the Wang–Xing–Safavi-Naini bound if and only if they are certain Steiner structures in [Formula: see text].

Bounds on Subspace Codes Based on Subspaces of Type(m,1)in Singular Linear Space

The Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codesn+l,M,d,(m,1)qbased on subspaces of type(m,1)in singular linear spaceFq(n+l)over finite fieldsFqare presented. Then, we prove that codes based on subspaces of type(m,1)in singular linear space attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures inFq(n+l).

On the Failing Cases of the Johnson Bound for Error-Correcting Codes

A central problem in coding theory is to determine $A_q(n,2e+1)$, the maximal cardinality of a $q$-ary code of length $n$ correcting up to $e$ errors. When $e$ is fixed and $n$ is large, the best upper bound for $A(n,2e+1)$ (the binary case) is the well-known Johnson bound from 1962. This however simply reduces to the sphere-packing bound if a Steiner system $S(e+1,2e+1,n)$ exists. Despite the fact that no such system is known whenever $e\geq 5$, they possibly exist for a set of values for $n$ with positive density. Therefore in these cases no non-trivial numerical upper bounds for $A(n,2e+1)$ are known. In this paper the author demonstrates a technique for upper-bounding $A_q(n,2e+1)$, which closes this gap in coding theory. The author extends his earlier work on the system of linear inequalities satisfied by the number of elements of certain codes lying in $k$-dimensional subspaces of the Hamming Space. The method suffices to give the first proof, that the difference between the sphere-packing bound and $A_q(n,2e+1)$ approaches infinity with increasing $n$ whenever $q$ and $e\geq 2$ are fixed. A similar result holds for $K_q(n,R)$, the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Moreover the author presents a new bound for $A(n,3)$ giving for instance $A(19,3)\leq 26168$.