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.

ON PLOTKIN-OPTIMAL CODES OVER FINITE FROBENIUS RINGS

We study the Plotkin bound for codes over a finite Frobenius ring R equipped with the homogeneous weight. We show that for codes meeting the Plotkin bound, the distribution on R induced by projection onto a coordinate has an interesting property. We present several constructions of codes meeting the Plotkin bound and of Plotkin-optimal codes. We also investigate the relationship between Butson–Hadamard matrices and codes over R meeting the Plotkin bound.

Some Remarks on the Plotkin Bound

In coding theory, Plotkin's upper bound on the maximal cadinality of a code with minimum distance at least $d$ is well known. He presented it for binary codes where Hamming and Lee metric coincide. After a brief discussion of the generalization to $q$-ary codes preserved with the Hamming metric, the application of the Plotkin bound to $q$-ary codes preserved with the Lee metric due to Wyner and Graham is improved.