By G. David Forney

Concatenation is a technique of establishing lengthy codes out of shorter ones; it makes an attempt to satisfy the matter of deciphering complexity by way of breaking the necessary computation into viable segments. We current theoretical and computational effects relating the potency and complexity of concatenated codes; the main theoretical effects are the following:

1. Concatenation of an arbitrarily huge variety of codes can yield a likelihood of errors that decreases exponentially with the over-all block size, whereas the interpreting complexity raises in basic terms algebraically; and

2. Concatenation of a finite variety of codes yields an errors exponent that's not as good as that possible with a unmarried level, yet is nonzero in any respect premiums under capacity.

Computations help those theoretical effects, and in addition provide perception into the connection among modulation and coding. This method illuminates the particular energy and usability of the category of Reed-Solomon codes. We provide an unique presentation in their constitution and homes, from which we derive the homes of all BCH codes; we be certain their weight distribution, and look at intimately the implementation in their deciphering set of rules, which we've prolonged to right either erasures and mistakes and feature differently more advantageous. 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 fitting interpreting innovations, which represent an engaging hybrid among interpreting and detection.

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**Example text**

For any assignment to the information places, so that J an arbitrary set of n - d + 1 places can be taken as an information set. It follows that Reed-Solomon codes are maximum and have minimum distance d. The complete dis- tribution of their weights has already been determined. As examples, RS codes on GF(4) have length 3 (or less). The code of all words sat- isfying the single equation fl + f 2 + f 3 = 0 (mo=0) has minimum distance 2. last symbol as the check symbol, we have f 3 = fl + f Taking the (we omit minus signs, since we are in a field of characteristic two), so that the code words are 000 101 aOa a20a 011 110 ala2 a 2 la Oaa laa2 aaO a2al Oa2a2 laZa aa21 aZaZ0 2 a + f3 a = 0 (m=O) has The code of all words satisfying f 1 +f2 +f3 = 0 andf l fa+f minimum distance 3.

Summary To summarize, there are for any kind of decoding two steps in which the number of computations is proportional to n. If we restrict ourselves to correcting deletions only, then there is no step in which the number of computations is proportional to more than d 2 Otherwise, reduction of the matrix M requires some computations that may be as 49 II_____I1I__I1II____lllllYII·61--··IU I-CI -_ - _ ·CI _ _ II large as d 3 . If we are doing general minimum-distance decoding, then we may have to repeat the computation d/2 times, which leads to a total number of computations proportional to d 4 .

An n where the a. are members of some field. The determinant of this matrix, D, also a member of the field, is a polynomial in the a i in which no a i appears to a power greater than n. Furthermore, since the determinant is zero if any two rows are the same, this polynomial must contain as factors a i - aj, all i D = D' j, so that I (a-a). ) contains each a to the n t h power, so that D' can only i>j 2 n 'a2 a n in this polynomial must be one, be a constant. ---.. ) -. 11_1- . Y·IIIILIIPII -CII^II(-l·P*I-_··I.

### Concatenated Codes by G. David Forney

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