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Asymptotically good codes have infinite trellis complexity

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2 Author(s)
Lafourcade, A. ; Coordinated Sci. Lab., Illinois Univ., Urbana, IL, USA ; Vardy, A.

The trellis complexity s(C) of a block code C is defined as the logarithm of the maximum number of states in the minimal trellis realization of the code. The parameter s(C) governs the complexity of maximum-likelihood decoding, and is considered a fundamental descriptive characteristic of the code in a number of recent works. We derive a new lower bound on s(C) which implies that asymptotically good codes have infinite trellis complexity. More precisely, for i⩾1 let Ci be a code over an alphabet of size q, of length ni, rate Ri, and minimum distance di. The infinite sequence of codes C, C2··· such that ni→∞ when i→∞ is said to be asymptotically good if both Ri and di/ni are bounded away from zero as i→∞. We prove that the complexity s(Ci) increases linearly with ni in any asymptotically good sequence of codes

Published in:

Information Theory, IEEE Transactions on  (Volume:41 ,  Issue: 2 )

Date of Publication:

Mar 1995

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