By G. David Forney
Concatenation is a technique of creating lengthy codes out of shorter ones; it makes an attempt to satisfy the matter of interpreting complexity by way of breaking the necessary computation into plausible segments. We current theoretical and computational effects referring to the potency and complexity of concatenated codes; the foremost theoretical effects are the following:
1. Concatenation of an arbitrarily huge variety of codes can yield a chance of errors that decreases exponentially with the over-all block size, whereas the interpreting complexity raises basically algebraically; and
2. Concatenation of a finite variety of codes yields an blunders exponent that's not so good as that possible with a unmarried degree, yet is nonzero in any respect charges less than capacity.
Computations help those theoretical effects, and likewise supply perception into the connection among modulation and coding. This process illuminates the precise energy and value of the category of Reed-Solomon codes. We provide an unique presentation in their constitution and houses, from which we derive the homes of all BCH codes; we make sure their weight distribution, and examine intimately the implementation in their deciphering set of rules, which we now have prolonged to right either erasures and mistakes and feature differently more desirable. We convey that on a very appropriate channel, RS codes can in achieving the functionality laid out in the coding theorem.
Finally, we current a generalization of using erasures in minimum-distance interpreting, and speak about the perfect interpreting ideas, which represent an engaging hybrid among deciphering and detection.
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F =0, or f I = - imp f.. imp Since p is arbitrary, every Clearly, this specifies a unique fp that solves the equation. set of n - 1 places is thus an information set, so that this code is a maximum code with length n, n - 1 information symbols, and minimum distance 2. a. Weight Distribution of Maximum Codes In general, the number of code words of given weight in a linear code is difficult or impossible to determine; for many codes even d, the minimum weight, is not accurately known. Surprisingly, determination of the weight distribution of a maximum code pre- sents no problems.
Furthermore, (p+y)P = pP + (P) p 1 + ... + (pPl) yp1 + by the binomial theorem; but every term but the first and last are multiplied by p, yP, therefore zero, and (+y)P = pP + yP, and y are elements of a field of charac- when teristic p. 3. 2 LINEAR CODES We know from the coding theorem that codes containing an exponentially large number of code words are required to achieve an exponentially low probability of error. Linear codes 4 ' 22 can contain such a great number of words, yet remain feasible to generate; they can facilitate minimum distance decoding, as we shall see.
Then there is an RS code on GF(q a ) with length n and minimum distance d. Since GF(q) is a subfield of GF(qa), there will be a certain subset of the code words in this code with all symbols in GF(q). The minimum distance between any two words in this subset must be at least as great as the minimum distance of the code, so that this subset can be taken as a code on GF(q) with length n 34 and minimum distance at least d. Any such subset is a BCH code. We shall call GF(q) the symbol field and GF(q a ) the locator field of the code.