Construction of multiple access channel codes based on hash property

Abstract

The aim of this paper is to introduce the construction of codes for a general discrete stationary memoryless multiple access channel based on the the notion of the hash property. Since an ensemble of sparse matrices has a hash property, we can use sparse matrices for code construction. Our approach has a potential advantage compared to the conventional random coding because it is expected that we can use some approximation algorithms by using the sparse structure of codes.

Channel Coding and Lossy Source Coding Using a Generator of Constrained Random Numbers

“…) → 0 as n → ∞, we have the fact that for any δ > 0 and sufficiently large n there are functions A ∈ A, and a vector c ∈ ImA satisfying (25) for all δ > 0 and sufficiently large n. Now, we prove (27) following the proof presented in [31]…”

“…we have the fact that for any δ > 0 and sufficiently large n there are functions A ∈ A, and a vector c ∈ ImA satisfying (25) for all δ > 0 and sufficiently large n. Now, we prove (27) following the proof presented in [31][14, Example 3.2.1]. Assume that µ Y n |X n is a channel with additive noise Z = {Y n − X n } ∞ n=1 .…”

“…In [21], the direct part of the channel coding theorem and the lossy source coding theorem for a discrete stationary memoryless channel/source are shown based on the hash property of an ensemble of functions, which is an extension of random bin coding [4], the set of all linear functions [6], and the two-universal class of hash functions [9]. In this paper, alternative general formulas for the channel capacity and rate-distortion region are introduced and the achievability of the proposed codes is proved based on a stronger version of hash property introduced in [22][23] [24] [25]. Since an ensemble of sparse matrices has a hash property, we can construct codes by using sparse matrices.…”

“…In[22][23][24][25], it is called the 'strong hash property.' Throughout this paper, we call it simply the 'hash property.…”

Channel Codes for Relayless Networks with General Message Access Structure

“…[30] [31][33] [36]; a similar idea is found in[9, Theorem 14.3] [40]. For each s P S, let us introduce a set C pnq s and two functions f s : Z n s Ñ C pnq s and g s : Z n s Ñ M pnq s , where the dependence of f s and g s on n is omitted.…”