Abstract
Linear codes over F q are considered for use in detecting and in correcting the additive errors in some subsetE of F nq (The most familiar example of such an error set E is the set of all n-tuples of Hamming weight at most t.) In this set-up, the basic averaging arguments for linear codes are reviewed with emphasis on the relation between the combinatorial and the information-theoretic viewpoint. The main theorems are (a correspondingly general version of) the Varshamov-Gilbert bound and a ‘random-coding’ bound on the probability of an ambiguous syndrome. These bounds are shown to result from applying the same elementary averaging argument to two different packing problems, viz., the combinatorial ‘sphere’ packing problem and the probabilistic ‘Shannon packing’. Some applications of the general bounds are indicated, e.g., hash functions and Euclidean-space codes, and the connection to Justesen-type constructions of asymptotically good codes is outlined.
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Loeliger, HA. (1994). On the Basic Averaging Arguments for Linear Codes. In: Blahut, R.E., Costello, D.J., Maurer, U., Mittelholzer, T. (eds) Communications and Cryptography. The Springer International Series in Engineering and Computer Science, vol 276. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2694-0_25
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DOI: https://doi.org/10.1007/978-1-4615-2694-0_25
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