The Viterbi decoding complexity of linear block codes

A given linear block code can be represented by many different trellises. In this paper, we will show that if these trellises are scored according to the complexity of implementing the Viterbi decoding algorithm on them, there is a uniquely optimal one, viz. The “Wolf (1978)-Massey (1978)-Muder (1988)” trellis. We will also introduce “minimal-span” generator matrices, which permit easy construction of WMM trellises

Practical codes for photon communication

In a recent paper, Pierce studied the problems of communicating at optical frequencies using photon-counting techniques, and concluded that "at low temperatures we encounter insuperable problems of encoding long before we approach [channel capacity]." In this paper it is shown that even assuming a noiseless model for photon communication for which capacity (measured in nats/photon) is infinite, it is unlikely that a signaling efficiency of even 10 nats/photon could be achieved practically. On the positive side, it is shown that pulse-position modulation plus Reed-Solomon coding yields practical results in the range of 2 to 3 nats/photon

Comments on 'A class of codes for axisymmetric channels and a problem from the additive theory of numbers' by Varshanov, R. R.

In the above paper [1], Varshamov considers discrete channels with q inputs and q outputs, q being an arbitrary integer

On the symmetry of good nonlinear codes

It is shown that there are arbitrarily long "good" (in the sense of Gilbert) binary block codes that are preserved under very large permutation groups. This result contrasts sharply with the properties of linear codes: it is conjectured that long cyclic codes are bad, and known that long affine-invariant codes are bad

Interleaved block codes for the photon channel

It is shown that interleavel binary block codes combined with pulse position modulation give the best practical coded systems yet devised for optical communication with photon detection. Linear block codes rather than convolutional codes are considered

RFI channels

A class of channel models is presented which exhibit varying burst error severity much like channels encountered in practice. An information-theoretic analysis of these channel models is made, and conclusions are drawn that may aid in the design of coded communication systems for realistic noisy channels

On the decoder error probability for Reed-Solomon codes

Upper bounds On the decoder error probability for Reed-Solomon codes are derived. By definition, "decoder error" occurs when the decoder finds a codeword other than the transitted codeword; this is in contrast to "decoder failure," which occurs when the decoder fails to find any codeword at all. These results imply, for example, that for a t error-correcting Reed-Solomon code of length q - 1 over GF(q), if more than t errors occur, the probability of decoder error is less than 1/t!

The trellis complexity of convolutional codes

Convolutional codes have a natural, regular, trellis structure that facilitates the implementation of Viterbi's algorithm. Linear block codes also have a natural, though not in general a regular, “minimal” trellis structure, which allows them to be decoded with a Viterbi-like algorithm. In both cases, the complexity of an unenhanced Viterbi decoding algorithm can be accurately estimated by the number of trellis edge symbols per encoded bit. It would therefore appear that we are in a good position to make a fair comparison of the Viterbi decoding complexity of block and convolutional codes. Unfortunately, however, this comparison is somewhat muddled by the fact that some convolutional codes, the punctured convolutional codes, are known to have trellis representations which are significantly less complex than the conventional trellis. In other words, the conventional trellis representation for a convolutional code may not be the “minimal” trellis representation. Thus ironically, we seem to know more about the minimal trellis representation for block than for convolutional codes. We provide a remedy, by developing a theory of minimal trellises for convolutional codes. This allows us to make a direct performance-complexity comparison for block and convolutional codes. A by-product of our work is an algorithm for choosing, from among all generator matrices for a given convolutional code, what we call a trellis-canonical generator matrix, from which the minimal trellis for the code can be directly constructed. Another by-product is that in the new theory, punctured convolutional codes no longer appear as a special class, but simply as high-rate convolutional codes whose trellis complexity is unexpectedly small

The Extended Invariant Factor Algorithm with Application to the Forney Analysis of Convolutional Codes

In his celebrated paper on the algebraic structure of convolutional codes, Forney showed that by using the invariant-factor theorem, one can transform an arbitrary polynomial generator matrix for an (n, k) convolutional code C into a basic (and ultimately a minimal) generator matrix for C. He also showed how to find a polynomial inverse for a basic generator matrix for C, and a basic generator matrix for the dual code C^⊥. In this paper, we will discuss efficient ways to do all these things. Our main tool is the “entended invariant factor algorithm,” which we introduce here