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19
Distributed Source Coding using Abelian Group Codes: Extracting Performance from Structure
"... In this work, we consider a distributed source coding problem with a joint distortion criterion depending on the sources and the reconstruction. This includes as a special case the problem of computing a function of the sources to within some distortion and also the classic SlepianWolf problem [12 ..."
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In this work, we consider a distributed source coding problem with a joint distortion criterion depending on the sources and the reconstruction. This includes as a special case the problem of computing a function of the sources to within some distortion and also the classic SlepianWolf problem [12], BergerTung problem [5], WynerZiv problem [4], YeungBerger problem [6] and the AhlswedeKornerWyner problem [3], [13]. While the prevalent trend in information theory has been to prove achievability results using Shannon’s random coding arguments, using structured random codes offer rate gains over unstructured random codes for many problems. Motivated by this, we present a new achievable ratedistortion region (an inner bound to the performance limit) for this problem for discrete memoryless sources based on “good” structured random nested codes built over abelian groups. We demonstrate rate gains for this problem over traditional coding schemes using random unstructured codes. For certain sources and distortion functions, the new rate region is strictly bigger than the BergerTung rate region, which has been the best known achievable rate region for this problem till now. Further, there is no known unstructured random coding scheme that achieves these rate gains. Achievable performance limits for singleuser source coding using abelian group codes are also obtained as parts of the proof of the main coding theorem. As a corollary, we also prove that nested linear codes achieve the Shannon ratedistortion bound in the singleuser setting. Note that while group codes retain some structure, they are more general than linear codes which can only be built over finite fields which are known to exist only for certain sizes.
Some structural properties of convolutional codes over rings
 IEEE Trans Inform Theor
, 1998
"... Abstract — Convolutional codes over rings have been motivated by phasemodulated 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 o ..."
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Cited by 9 (0 self)
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Abstract — Convolutional codes over rings have been motivated by phasemodulated 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. I.
Minimal Gröbner bases and the predictable leading monomial property
, 2009
"... In this paper we focus on Gröbner bases over rings for the univariate case. We identify a useful property of minimal Gröbner bases, that we call the “predictable leading monomial (PLM) property”. The property is stronger than “row reducedness ” and is crucial in a range of applications. The first pa ..."
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Cited by 6 (6 self)
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In this paper we focus on Gröbner bases over rings for the univariate case. We identify a useful property of minimal Gröbner bases, that we call the “predictable leading monomial (PLM) property”. The property is stronger than “row reducedness ” and is crucial in a range of applications. The first part of the paper is tutorial in outlining how the PLM property enables straightforward solutions to classical realization problems of linear systems over fields. In the second part of the paper we use the ideas of [20] on polynomial matrices over the finite ring Zpr (with p a prime integer and r a positive integer) in the more general setting of Gröbner bases and introduce the notion of “Gröbner pbasis ” to achieve a predictable leading monomial property over Zpr. This theory finds applications in error control coding over Zpr. Through this approach we are extending the ideas of [20] to a more general context where the user chooses an ordering of polynomial vectors. 1
The "Art of Trellis Decoding" is Computationally Hard  for Large Fields
 IEEE TRANS. INFORM. THEORY
, 1998
"... The problem of minimizing the trellis complexity of a code by coordinate permutation is studied. Three measures of trellis complexity are considered: the total number of states, the total number of edges, and the maximum state complexity of the trellis. The problem is proven NPhard for all three ..."
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Cited by 3 (0 self)
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The problem of minimizing the trellis complexity of a code by coordinate permutation is studied. Three measures of trellis complexity are considered: the total number of states, the total number of edges, and the maximum state complexity of the trellis. The problem is proven NPhard for all three measures, provided the field over which the code is specified is not fixed. We leave open the problem of dealing with the case of a fixed field, in particular GF 2).
Minimal trellis construction for finite support convolutional ring codes
 Coding Theory and Applications (ICMCTA), LN in Computer Science 5228
, 2008
"... Abstract. We address the concept of “minimal polynomial encoder ” for finite support linear convolutional codes over Zpr. These codes can be interpreted as polynomial modules which enables us to apply results from the 2007paper [8] to introduce the notions of “pencoder ” and “minimal pencoder”. H ..."
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Abstract. We address the concept of “minimal polynomial encoder ” for finite support linear convolutional codes over Zpr. These codes can be interpreted as polynomial modules which enables us to apply results from the 2007paper [8] to introduce the notions of “pencoder ” and “minimal pencoder”. Here the latter notion is the ring analogon of a row reduced polynomial encoder from the field case. We show how to construct a minimal trellis representation of a delayfree finite support convolutional code from a minimal pencoder. We express its number of trellis states in terms of a degree invariant of the code. The latter expression generalizes the wellknown expression in terms of the degree of a delayfree finite support convolutional code over a field to the ring case. The results are also applicable to block trellis realization of polynomial block codes over r, such as CRC codes over Zpr.
Blockcoded PSK modulation using twolevel group codes over dihedral groups
 IEEE TRANS. INFORM. THEORY
, 1998
"... A length n group code over a group G is a subgroup of G under componentwise group operation. Group codes over dihedral groups hw, with Pw elements, that are twolevel constructible using a binary code and a code over w residue class integer ring modulo w, as component codes are studied for arbitr ..."
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Cited by 2 (1 self)
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A length n group code over a group G is a subgroup of G under componentwise group operation. Group codes over dihedral groups hw, with Pw elements, that are twolevel constructible using a binary code and a code over w residue class integer ring modulo w, as component codes are studied for arbitrary w. A set of necessary and sufficient conditions on the component codes for the twolevel construction to result in a group code over hw are obtained. The conditions differ for w odd and even. Using twolevel group codes over hw as label codes, performance of blockcoded modulation scheme is discussed under all possible matched labelings of PwAPSK and PwSPSK (asymmetric and symmetric PSK) signal sets in terms of the minimum squared Euclidean distance. Matched labelings that lead to Automorphic Euclidean Distance Equivalent codes are identified. It is shown that depending upon the ratio of Hamming distances of the component codes some labelings perform better than other. The best labeling is identified under a set of restrictive conditions. Finally, conditions on the component codes for phase rotational invariance properties of the signal space codes are discussed.
Minimal trellis construction from convolutional ring encoders
, 801
"... The paper addresses minimality of encoders for basic convolutional codes over Zpr by using a recently developed concept of row reducedness for polynomial matrices over Zpr. It is known in the literature that the McMillan degree of a basic encoder is an upper bound for the minimum number of trellis s ..."
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The paper addresses minimality of encoders for basic convolutional codes over Zpr by using a recently developed concept of row reducedness for polynomial matrices over Zpr. It is known in the literature that the McMillan degree of a basic encoder is an upper bound for the minimum number of trellis states, but a general expression is missing. This open problem is solved in this paper. An expanded type of polynomial encoder is introduced, called “pencoder”, whose rows are required to be a pgenerator sequence. The latter property enables the working of fundamental linear algebraic properties, such as linear independence. It is shown that for any basic convolutional code a particular type of pencoder can be constructed that is the ring analog of a canonical encoder from the field case. The open problem of constructing a minimal trellis representation of the code is then solved and the minimum number of trellis states is expressed in terms of an algebraic degree invariant of the code. In the literature this problem was only solved for the restrictive case where the code admits a basic row reduced encoder. The obtained results hold for the classical setup of left compact support convolutional codes. It is also shown how they are extended to finite support convolutional codes. 1
Published: 19980101
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