Singular Manual: Codes and the decoding problem

| ### C.8.1 Codes and the decoding problem #### Codes Let 797#797 be a field with 798#798 elements. A linear code 78#78 is a linear subspace of 799#799 endowed with the Hamming metric. Hamming distance between x,y 800#800. Hamming weight of x 801#801. Minimum distance of the code 802#802. The code 78#78 of dimension 280#280 and minimum distance 171#171 is denoted as 803#803. A matrix 189#189 whose rows are the base vectors of 78#78 is the generator matrix. A matrix 804#804 with the property 805#805 is the check matrix. #### Cyclic codes The code 78#78 is cyclic, if for every codeword 806#806 in 78#78 its cyclic shift 807#807 is again a codeword in 78#78. When working with cyclic codes, vectors are usually presented as polynomials. So 808#808 is represented by the polynomial 809#809 with 810#810, more precisely 811#811 is an element of the factor ring 812#812. Cyclic codes over 797#797 of length 17#17 correspond one-to-one to ideals in this factor ring. We assume for cyclic codes that 813#813. Let 814#814 be the splitting field of 815#815 over 797#797. Then 708#708 has a primitive 17#17-th root of unity which will be denoted by 4#4. A cyclic code is uniquely given by a defining set 816#816 which is a subset of 817#817 such that 818#818 A cyclic code has several defining sets. #### Decoding problem Complete decoding: Given 819#819 and a code 820#820, so that 41#41 is at distance 821#821 from the code, find 822#822. Bounded up to half the minimum distance: With the additional assumption 823#823, a codeword with the above property is unique. #### Decoding via systems solving One distinguishes between two concepts: Generic decoding: Solve some system 824#824 and obtain some "closed" formulas 825#825. Evaluating these formulas at data specific to a received word 826#826 should yield a solution to the decoding problem. For example for 827#827. The roots of 828#828 yield error positions, see the section on the general error-locator polynomial. Online decoding: Solve some system 829#829. The solutions should solve the decoding problem. #### Computational effort Generic decoding. Here, preprocessing is very hard, whereas decoding is relatively simple (if the formulas are sparse). Online decoding. In this case, decoding is the hard part.

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