Some structural properties of convolutional codes over rings

Abstract

Abstract-Convolutional codes over rings have been motivated by phase-modulated signals. Some structural properties of the generator matrices of such codes are presented. Successively stronger notions of invertibility of generator matrices are studied, and a new condition for a convolutional code over a ring to be systematic is given and shown to be equivalent to a condition given by Massey and Mittelholzer. It is shown that a generator matrix that can be decomposed into a direct sum is basic, minimal, and noncatastrophic if and only if all generator matrices for the constituent codes are basic, minimal, and noncatastrophic, respectively. It is also shown that if a systematic generator matrix can be decomposed into a direct sum, then all generator matrices of the constituent codes are systematic, but that the converse does not hold. Some results on convolutional codes over p are obtained.Index Terms-Convolutional codes over rings, direct sum decomposition of rings, proper convolutional codes, systematic convolutional codes.

Right/Left inverses of polynomial matrices for control, communications and vision

“…Quite similar families of applications include distortionless (or perfect) reception in error-control coding [7], [8], [9], [10], [11], [27], [28], perfect precoding and equalization [29], [30], [31], [32], [33], [34], [35], perfect signal reconstruction, including perfect reconstruction filter banks [12], [13], [14], [36], [37], [38] and perfect deconvolution [39], [40], also in the problem of image recovery [41], [42]. On the other hand, the problem of perfect input reconstruction that has been solved employing the state-space approach [43] could as well be tackled and resolved on the basis of the rational matrix system description.…”

A study on new right/left inverses of nonsquare polynomial matrices

“…Quite similar families of applications include distortionless (or perfect) reception in error-control coding (Fornasini and Pinto, 2004;Forney, 1991;GluesingLuerssen et al, 2006;Johannesson and Zigangirov, 1999;Johannesson et al, 1998;Lin and Costello, 2004;Lu et al, 2005;Moon, 2005), perfect precoding and equalization (Boche and Pohl, 2006;Guo and Levy, 2004;Kim and Park, 2004;Tidestav et al, 1999;Wahls and Boche, 2007;Wahls et al, 2009;Xia et al, 2001), perfect signal reconstruction, including perfect reconstruction filter banks (Bernardini and Rinaldo, 2006;Gan and Ling, 2008;Law et al, 2009;Quevedo et al, 2009;Vaidyanathan and Chen, 1995;Zhang and Makur, 2009) and perfect deconvolution (Inouye and Tanebe, 2000;Tuncer, 2004), also in the problem of image recovery (Castella and Pesquet, 2004;Harikumar and Bresler, 1999a;1999b). On the other hand, the problem of perfect input reconstruction that has been solved employing the statespace approach (Edelmayer et al, 2004) could as well be tackled and resolved on the basis of the rational matrix system description.…”

Two 16-state, rate R=2/4 trellis codes whose free distances meet the Heller bound

“…A rate R = b=c convolutional code over a ring is systematic if and only if it has a generator matrix that has a b 2b subdeterminant which is a unit in the ring of realizable functions [6].…”