| ### C.8.5 Decoding method based on quadratic equations #### Preliminary definitions Let 941#941 be a basis of 799#799 and let 195#195 be the 234#234 matrix with 941#941 as rows. The unknown syndrome 942#942 of a word 834#834 w.r.t 195#195 is the column vector 943#943 with entries 944#944 for 945#945. For two vectors 946#946 define 947#947. Then 948#948 is a linear combination of 941#941, so there are constants 949#949 such that 950#950 The elements 949#949 are the structure constants of the basis 941#941. Let 951#951 be the 952#952 matrix with 953#953 as rows (954#954). Then 941#941 is an ordered MDS basis and 195#195 an MDS matrix if all the 955#955 submatrices of 951#951 have rank 177#177 for all 956#956. #### Expressing known syndromes Let 78#78 be an 797#797-linear code with parameters 803#803. W.l.o.g 957#957. 804#804 is a check matrix of 78#78. Let 958#958 be the rows of 804#804. One can express 959#959 with some 960#960. In other words 961#961 where 190#190 is the 962#962 matrix with entries 963#963. Let 832#832 be a received word with 833#833 and 834#834 an error vector. The syndromes of 826#826 and 834#834 w.r.t 804#804 are equal and known: 964#964 They can be expressed in the unknown syndromes of 834#834 w.r.t 195#195: 965#965 since 959#959 and 966#966. #### Constructing the system Let 195#195 be an MDS matrix with structure constants 967#967. Define 968#968 in the variables 969#969 by 970#970 The ideal 971#971 in 972#972 is generated by 973#973 The ideal 974#974 in 975#975 is generated by 976#976 Let 977#977 be the ideal in 975#975 generated by 971#971 and 974#974. #### Main theorem Let 195#195 be an MDS matrix with structure constants 967#967. Let 804#804 be a check matrix of the code 78#78 such that 961#961 as above. Let 832#832 be a received word with 833#833 the codeword sent and 834#834 the error vector. Suppose that 978#978 and 979#979. Let 501#501 be the smallest positive integer such that 977#977 has a solution 980#980 over the algebraic closure of 797#797. Then 981#981 and the solution is unique and of multiplicity one satisfying 982#982. the reduced Gröbner basis 189#189 for the ideal 977#977 w.r.t any monomial ordering is 983#983 984#984 where 985#985 is the unique solution. For an example see `sysQE` in decodegb_lib. More on this method can be found in [BP2008a].

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