Bounds on algebraic code capacities for noisy channels. I

IEEE Trans. Inform. Theory

2015

In this paper, we study the asymptotic performance of Abelian group codes for the the channel coding problem for arbitrary discrete (finite alphabet) memoryless channels as well as the lossy source coding problem for arbitrary discrete (finite alphabet) memoryless sources. For the channel coding problem, we find the capacity characterized in a single-letter information-theoretic form. This simplifies to the symmetric capacity of the channel when the underlying group is a field. For the source coding problem, we derive the achievable rate-distortion function that is characterized in a single-letter information-theoretic form. When the underlying group is a field, it simplifies to the symmetric rate-distortion function. We give several illustrative examples. Due to the non-symmetric nature of the sources and channels considered, our analysis uses a synergy of information-theoretic and group-theoretic tools.

Section: Introductionmentioning

confidence: 99%

“…In [6], the capacity of group codes for certain classes of channels has been computed. Further results on the capacity of group codes were established in [7], [8]. The capacity of group codes over a class of channels exhibiting symmetries with respect to the action of a finite Abelian group has been investigated in [11].…”

Section: Introductionmentioning

confidence: 99%

Abelian Group Codes for Channel Coding and Source Coding

Sahebi

IEEE Trans. Inform. Theory

2015

“…The capacity of linear codes has been studied in [2]. We show that for the case of linear codes over F q , the upper and lower bounds are tight and are equal to the capacity given in [2]. Let C be a group code over the field F q for some prime number q.…”

Section: A Linear Codesmentioning

confidence: 97%

On the capacity of Abelian group codes over discrete memoryless channels

Sahebi

2011 IEEE International Symposium on Information Theory Proceedings

2011

Abstract-For most discrete memoryless channels, there does not exist a linear code which uses all of the channel's input symbols. Therefore, linearity of the code for such channels is a very restrictive condition and there should be a loosening of the algebraic structure of the code to a degree that the code can admit any channel input alphabet. For any channel input alphabet size, there always exists an Abelian group structure defined on the alphabet. We investigate the capacity of Abelian group codes over discrete memoryless channels and provide lower and upper bounds on the capacity.

“…. Therefore, if we were to communicate over a PTP-STx (S, W S , X , κ, Y, W Y |XS ) using codes over an Abelian Z p r −group V = Z p r and we constrained the bins to be closed under group addition, then the test channel p V SXY ∈ D(τ ) yields an achievable rate log |V|−H(V |Y )−(I V s (V ; S)) = 18 The interested reader is referred to [37], [38], [39] for early work on rates achievable using group codes for point-to-point channels. [40] provides bounds on rates achievable using Abelian group codes for point-to-point source and channel coding problems.…”

Section: Stage Iii: Achievable Rate Region Using Codes Over Abementioning

confidence: 99%

An Achievable Rate Region Based on Coset Codes for Multiple Access Channel With States

Padakandla

IEEE Trans. Inform. Theory

2017

We prove that the ensemble the nested coset codes built on finite fields achieves the capacity of arbitrary discrete memoryless point-to-point channels. Exploiting it's algebraic structure, we develop a coding technique for communication over general discrete multiple access channel with channel state information distributed at the transmitters. We build an algebraic coding framework for this problem using the ensemble of Abelian group codes and thereby derive a new achievable rate region. We identify non-additive and non-symmteric examples for which the proposed achievable rate region is strictly larger than the one achievable using random unstructured codes.