By G. David Forney
Concatenation is a technique of establishing lengthy codes out of shorter ones; it makes an attempt to fulfill the matter of deciphering complexity by means of breaking the necessary computation into practicable segments. We current theoretical and computational effects concerning the potency and complexity of concatenated codes; the most important theoretical effects are the following:
1. Concatenation of an arbitrarily huge variety of codes can yield a chance of blunders that decreases exponentially with the over-all block size, whereas the interpreting complexity raises purely algebraically; and
2. Concatenation of a finite variety of codes yields an blunders exponent that's not as good as that possible with a unmarried degree, yet is nonzero in any respect charges less than capacity.
Computations aid those theoretical effects, and likewise provide perception into the connection among modulation and coding. This method illuminates the designated strength 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 verify their weight distribution, and view intimately the implementation in their deciphering set of rules, which now we have prolonged to right either erasures and error and feature differently more advantageous. We exhibit that on a very compatible channel, RS codes can in attaining the functionality laid out in the coding theorem.
Finally, we current a generalization of using erasures in minimum-distance interpreting, and speak about the correct deciphering suggestions, which represent a fascinating hybrid among interpreting and detection.
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Since what we are really interested in is the relationship between probability of error and complexity for a given rate, we can eliminate L from these two relationships in this way: if complexity is algebraic in L while Pr(e) is exponential in L, Pr(e) is exponential in complexity, while if both complexity and Pr(e) are exponential in L, Pr(e) is only algebraic in complexity. For example, the coding theorem uses maximum-likelihood decoding of block codes of length N to achieve error probability Pr(e) e NE(R).
EFFICIENCY AND COMPLEXITY We shall now collect our major theoretical results on concatenated codes. We find that by concatenating we can achieve exponential decrease of probability of error with over-all block length, with only an algebraic increase in decoding complexity, for all rates below capacity; on an ideal superchannel with a great many inputs, Reed-Solomon codes can match the performance specified by the coding theorem; and with two stages of concatenation we can get a nonzero error exponent at all rates below capacity, although this exponent will be less than the unconcatenated exponent.
42 are exactly the r distinct Eqs. 2, since Eqs. 2 are the distinct equations in Eqs. 1. Thus fq fq +l Eqs. 41 for the same information symbols fi, i E S, as did fbfb+l,. were shown to be the unique solution to Eqs. 41. fq 'fb+r-I' solve which Therefore fq+i = fb+i; but the elements a of GF(q ) which satisfy pq = P are precisely the elements of GF(q) , so that the fb+i are elements of GF(q). n-r Thus the code has an information set of n - r symbols, and therefore there are q code words. 36 _ We remark that any set of r places whose locators can be represented as r consecutive powers of some field element is thus a check set, and the remaining n - r an information set.
Concatenated Codes by G. David Forney