Source and channel rate allocation for channel codes satisfying the Gilbert-Varshamov or Tsfasman-Vladut-Zink bounds

Abstract

We derive bounds for optimal rate allocation between source and channel coding for linear channel codes that meet the Gilbert-Varshamov or Tsfasman-Vlȃduţ-Zink bounds. Formulas giving the high resolution vector quantizer distortion of these systems are also derived. In addition, we give bounds on how far below channel capacity the transmission rate should be for a given delay constraint. The bounds obtained depend on the relationship between channel code rate and relative minimum distance guaranteed by the Gilbert-Varshamov bound, and do not require sophisticated decoding beyond the error correction limit. We demonstrate that the end-to-end mean-squared error decays exponentially fast as a function of the overall transmission rate, which need not be the case for certain well-known structured codes such as Hamming codes.

Performance of quantizers on noisy channels using structured families of codes

“…However, those results assume the existence of optimal channel codes, namely, those described in Shannon's channel coding theorem using random coding arguments. Similar techniques were used to generalize the results of [5] to Gaussian channels [6] and to certain algebraic-geometry codes [7]. Hence, the results in [5]- [7] are existence constructions and do not necessarily correspond to achievable performance based on the best presently known implementable channel codes.…”

“…Similar techniques were used to generalize the results of [5] to Gaussian channels [6] and to certain algebraic-geometry codes [7]. Hence, the results in [5]- [7] are existence constructions and do not necessarily correspond to achievable performance based on the best presently known implementable channel codes. There is thus motivation to find a high resolution theory for quantization with a noisy channel, using families of structured algebraic channel codes.…”

“…A key assumption in [5], [7] is that by keeping the channel code rate fixed (below capacity) while increasing the overall transmission rate , the probability of decoding error can decay to zero exponentially fast as a function of . This assumption is valid for "Shannon-optimal" codes and more generally for asymptotically good codes, but most known structured fam-0018-9448/00$10.00 © 2000 IEEE ilies of channel codes (e.g., Hamming, Bose-Chaudhuri-Hocquenghem (BCH), Reed-Muller) do not have this property.…”

“…An important design parameter is the allocation of the available transmission rate between source and channel coding. Tight upper and lower bounds on the optimal tradeoff between source and channel coding are known for certain codes and channels and th-power distortion measures [5]- [7], [4]. These results exploit the fact that the distortion contributions of optimal source coding and optimal channel coding decay exponentially fast as functions of the overall transmission rate.…”

“…These results exploit the fact that the distortion contributions of optimal source coding and optimal channel coding decay exponentially fast as functions of the overall transmission rate. The source coder is taken to be a "good" vector quantizer (one that obeys Zador's decay rate) in [5]- [7], [4], and index assignment randomization is used. In both [5] and [6], the channel codes are assumed to have exponentially decaying error probabilities achieving the expurgated error exponent for the given channel (a binary symmetric channel in [5] and an additive white Gaussian noise channel in [6]).…”