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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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