Q-ary Constant-weight Codes Research Articles

A new construction for q-ary constant weight codes

For a given prime power q, a new method for constructing q-ary constant weight codes is presented, and then several classes of q-ary constant weight codes are given by using the trace map over finite fields. Especially, for q=2, some of these codes are optimal, and then the maximal possible sizes of two classes of binary constant weight codes are determined.

On m-Near-Resolvable Block Designs and q-ary Constant-Weight Codes

We introduce m-near-resolvable block designs. We establish a correspondence between such block designs and a subclass of (optimal equidistant) q-ary constant-weight codes meeting the Johnson bound. We present constructions of m-near-resolvable block designs, in particular based on Steiner systems and super-simple t-designs.

On m-Quasi-resolvable Block Designs and q-ary Constant-Weight Codes

On m-Quasi-resolvable Block Designs and q-ary Constant-Weight Codes

Erratum to: “Remark on balanced incomplete block designs, near-resolvable block designs, and q-ary constant-weight codes”

The acknowledgment (footnote to the title of the paper) should read as follows: The research was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50- 00150.

Remark on balanced incomplete block designs, near-resolvable block designs, and q-ary constant-weight codes

Remark on balanced incomplete block designs, near-resolvable block designs, and q-ary constant-weight codes

Completely reducible super-simple designs with block size five and index two

Complete reducible super-simple (CRSS) designs are closely related to $$q$$q-ary constant weight codes. A $$(v,k,\lambda )$$(v,k,?)-CRSS design is just an optimal $$(v,2(k-1),k)_{\lambda +1}$$(v,2(k-1),k)?+1 code. In this paper, we mainly investigate the existence of a $$(v,5,2)$$(v,5,2)-CRSS design and show that such a design exists if and only if $$v\equiv 1,5\pmod {20}$$v?1,5(mod20) and $$v\ge 21$$v?21, except possibly when $$v = 25$$v=25. Using this result, we determine the maximum size of an $$(n,8,5)_3$$(n,8,5)3 code for all $$n\equiv 0,1,4,5 \pmod {20}$$n?0,1,4,5(mod20) with the only length $$n=25$$n=25 unsettled. In addition, we also construct super-simple $$(v,5,3)$$(v,5,3)-BIBDs for $$v=45,65$$v=45,65.

Completely reducible super-simple designs with block size four and related super-simple packings

A design is said to be super-simple if the intersection of any two blocks has at most two elements. A super-simple design $${\mathcal{D}}$$ with point set X, block set $${\mathcal{B}}$$ and index ? is called completely reducible super-simple (CRSS), if its block set $${\mathcal{B}}$$ can be written as $${\mathcal{B}=\bigcup_{i=1}^{\lambda} \mathcal{B}_i}$$ , such that $${\mathcal{B}_i}$$ forms the block set of a design with index unity but having the same parameters as $${\mathcal{D}}$$ for each 1 ? i ? ?. It is easy to see, the existence of CRSS designs with index ? implies that of CRSS designs with index i for 1 ? i ? ?. CRSS designs are closely related to q-ary constant weight codes (CWCs). A (v, 4, q)-CRSS design is just an optimal (v, 6, 4) q+1 code. On the other hand, CRSS group divisible designs (CRSSGDDs) can be used to construct q-ary group divisible codes (GDCs), which have been proved useful in the constructions of q-ary CWCs. In this paper, we mainly investigate the existence of CRSS designs. Three neat results are obtained as follows. Firstly, we determine completely the spectrum for a (v, 4, 3)-CRSS design. As a consequence, a class of new optimal (v, 6, 4)4 codes is obtained. Secondly, we give a general construction for (4, 4)-CRSSGDDs with skew Room frames, and prove that the necessary conditions for the existence of a (4, 2)-CRSSGDD of type g u are also sufficient except definitely for $${(g,u)\in \{(2,4),(3,4),(6,4)\}}$$ . Finally, we consider the related optimal super-simple (v, 4, 2)-packings and show that such designs exist for all v ? 4 except definitely for $${v\in \{4,5,6,9\}}$$ .

Optimal Ternary Constant-Weight Codes of Weight Four and Distance Six

Recently, Chee and Ling (?Constructions for q-ary constant-weight codes?, IEEE Trans. Inf. Theory, vol. 53, no. 1, 135-146, Jan. 2007 ) introduced a new combinatorial construction for q-ary constant-weight codes which reveals a close connection between q-ary constant-weight codes and sets of pairwise disjoint combinatorial designs. In this paper, we study the problem of constructing optimal ternary constant-weight codes with Hamming weight four and minimum distance six using this approach. The construction here exploits completely reducible super simple designs and group divisible codes. The problem is solved leaving only two cases undetermined. Previously, the sizes of constant-weight codes of weight four and distance six were known only for those of length no greater than 10.

Undetected error probability of q-ary constant weight codes

In this paper, we introduce a new combinatorial invariant called q-binomial moment for q-ary constant weight codes. We derive a lower bound on the q-binomial moments and introduce a new combinatorial structure called generalized (s, t)-designs which could achieve the lower bounds. Moreover, we employ the q-binomial moments to study the undetected error probability of q-ary constant weight codes. A lower bound on the undetected error probability for q-ary constant weight codes is obtained. This lower bound extends and unifies the related results of Abdel-Ghaffar for q-ary codes and Xia-Fu-Ling for binary constant weight codes. Finally, some q-ary constant weight codes which achieve the lower bounds are found.

Constructions for $q$-Ary Constant-Weight Codes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper introduces a new combinatorial construction for <emphasis><formula formulatype="inline"><tex>$q$</tex></formula></emphasis>-ary constant-weight codes which yields several families of optimal codes and asymptotically optimal codes. The construction reveals intimate connection between <emphasis><formula formulatype="inline"><tex>$q$</tex></formula></emphasis>-ary constant-weight codes and sets of pairwise disjoint combinatorial designs of various types. </para>