Abstract
In general, the linear codes discussed in the last chapter require complicated logic circuitry and a large storage memory for both the encoding and decoding processes when the value of n-k is large. To illustrate for standard-array decoding of an (n, k) binary linear code requires the storage of 2n−k cosets and 2k words each of n bits, e.g. for the (32,6) binary linear code the storage to 232 × 32 = 234 × 8 ≈ 6Gbytes. To reduce this complexity, syndrome decoding was introduced. But one still has to store 2n−k coset leaders and 2n−k syndromes, a total of 2n−k × n + 2n−k × (n − k)bits. Again for the (32,6) binary linear code, the storage requirement is 226 × 32 + 226 × 26 ≈ 500Mbytes. Obviously, such coding costs are too high to be acceptable for any application of this simple (32,6) binary linear code. Therefore, for a simpler encoding and decoding some additional properties of a code other than the linear structure need to be used. The linear cyclic codes, discussed in this chapter, can be implemented relatively easily and possess as well a great deal of well-understood mathematical structure.
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Reed, I.S., Chen, X. (1999). Linear Cyclic Codes. In: Error-Control Coding for Data Networks. The Springer International Series in Engineering and Computer Science, vol 508. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5005-1_4
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