Perfect Codes and Related Structures

Abstract

A de Bruijn arrays code is a set of r × s binary doubly-periodic arrays such that each binary n × m matrix is contained exactly once as a window in one of the arrays. Such a set of arrays can be viewed as a two-dimensional generalization of a perfect factor in the de Bruijn graph. Necessary conditions for the existence of such arrays are given. Several direct constructions and recursive constructions for such arrays are given. A framework for a theory of two-dimensional feedback shift register which is akin to (one-dimensional) feedback shift registers is suggested.

Multi-User Linearly-Separable Distributed Computing

“…For the case where (K, N ) accepts a perfect code, we know (cf. [35]) that µ τ = 1 for which the upper bounds in ( 26) and ( 30) match the corresponding upper bounds in (31).…”

“…For this reason we are looking for codes (whose parity check matrix will be used as the decoding matrix D) that indeed minimize ρ and increase µ τ for a fixed τ . This special and rare property sought in C D is indeed attributed to the well-known class of perfect codes [35].…”

“…Proof. The proof is direct by noting that quasi-perfect codes have µ τ > 1 − N τ +1 (q − 1) τ +1 q −K and τ = ρ − 1 [35]. Remark 2.…”

“…To the best of our understanding, this is the first time that perfect codes (and the closely related quasi-perfect codes) have been associated with distributed computing and the equivalent problem of matrix factorization. The derived novel bounds on the cumulative computational cost as well as on the computational delay of a multi-user linearly-decomposable system capture the importance of the packing density as well as the packing and covering radius (defined in [35]) of a code whose parity-check matrix is our communication-and-computing matrix D. . Paper Organization and Notation: Section II introduces the system model, Section III formulates the problem, Section IV presents the main results, and finally, Section V concludes.…”