On perfect integer codes

Abstract

A general construction for perfect integer codes is provided, which allows to efficiently compute such codes. The method is applied to investigate in detail the special special error set {±1, ±a, ±b, ±c, } interesting for single error correction of peak shifts and codes defined on lattices.

Group Factorizations and Information Theory

“…He utilized a packing of {−1, 1} in C (10), presented the equivalent of a splitting of C(5) ⊕ C(5) by {−1, 1}. However this paper did not lead to subsequent investigation of crosses or semicrosses in Euclidean space..." "... Golomb and Welch showed that the (1, n)-cross tiles R n .…”

“…In this case the syndromes of a perfect singleerror correcting code form a factorization of the multiplicative group Z * p . We shall briefly sketch this connection and refer to the paper [10] for a detailed discussion.…”

“…Integer codes are a useful tool for single-error-correction, for instance, for peak shift correction for RLL codes [7], distortion of single components [8], codes on lattices [9] or permutation of components [10].…”

“…The analysis of the existence and construction of perfect integer codes is usually reduced to groups of prime orders [11], [13], [14], since constructions for composite m can be obtained from those for prime m. An efficient way to construct perfect integer codes was described in [13]. Here, the elements of E are expressed as powers of a generator g, namely a i = g µ i for i = 0, .…”

Integer SEC-DED codes for low power communications

“…On the other hand, in coding theory it is well known that the use of integer arithmetic provides a much simpler mechanism of error control, which is also confirmed by the several code constructions presented in [4][5][6]. However, although these codes are characterized by low coding/decoding complexity, they have the possibility to correct only specific types of errors (i.e.…”