Decoding and optimized implementation of SECDED codes over GF(q)
A plurality of columns for a check matrix that implements a distance d linear error correcting code are populated by providing a set of vectors from which to populate the columns, and applying to the set of vectors a filter operation that reduces the set by eliminating therefrom all vectors that would, if used to populate the columns, prevent the check matrix from satisfying a column-wise linear independence requirement associated with check matrices of distance d linear codes. One of the vectors from the reduced set may then be selected to populate one of the columns. The filtering and selecting repeats iteratively until either all of the columns are populated or the number of currently unpopulated columns exceeds the number of vectors in the reduced set. Columns for the check matrix may be processed to reduce the amount of logic needed to implement the check matrix in circuit logic.
1. A method of populating a plurality of columns for a check matrix that implements a distance d linear code, comprising:
providing a set of vectors from which to populate said columns;
applying to said set of vectors a filter operation that reduces said set by eliminating from said set all of said vectors that would, if used to populate said columns, prevent the check matrix from satisfying a column-wise linear independence requirement associated with check matrices of distance d linear codes and, thereafter, selecting from said set one of said vectors to populate one of said columns; and
iteratively repeating said applying and thereafter selecting, until occurrence of one of
population of all said columns, and
reduction of said set to a total number of said vectors less than a total number of said columns that are currently unpopulated.
2. The method of claim 1 , wherein said selecting includes selecting said one vector randomly.
3. The method of claim 1 , wherein said providing a set includes providing an initial set that contains all possible vectors from which said columns can be populated.
4. The method of claim 3 , wherein said applying said filter operation to said initial set includes eliminating from said initial set every vector that is a linear combination that includes any column of the identity matrix.
5. The method of claim 4 , wherein said applying said filter operation during said iteratively repeating includes, during an nth iteration of said applying and thereafter selecting, eliminating from the associated set every vector that is a linear combination that includes one of
the vector selected during an immediately preceding iteration of said applying and thereafter selecting,
any vector selected during any other preceding iteration of said applying and thereafter selecting, and
any column of the identity matrix.
6. The method of claim 5 , wherein said selecting includes selecting said one vector randomly.
7. The method of claim 1 , wherein said applying said filter operation during said iteratively repeating includes, during an nth iteration of said applying and thereafter selecting, eliminating from the associated set every vector that is a linear combination that includes one of
the vector selected during an immediately preceding iteration of said applying and thereafter selecting,
any vector selected during any other preceding iteration of said applying and thereafter selecting, and
any column of the identity matrix.
8. The method of claim 7 , wherein said selecting includes selecting said one vector randomly.
9. The method of claim 1 , including, after all of said columns are populated, processing said vectors of at least one of said populated columns to increase a number of zeros contained therein, and thereafter selecting less than all of said populated columns for use in the check matrix, wherein the associated vectors of all of said populated columns that are selected respectively contain a number of zeros that is less than or equal to a number of zeros contained in any said vector associated with the remainder of said populated columns.
10. An apparatus for populating a plurality of columns for a check matrix that implements a distance d linear code, comprising:
an input for providing a set of vectors from which to populate said columns;
a filter coupled to said input and said column and configured for
applying to said set of vectors a filter operation that reduces said set by eliminating from said set all of said vectors that would, if used to populate said columns, prevent the check matrix from satisfying a column-wise linear independence requirement associated with check matrices of distance d linear codes and,
thereafter, selecting from said set one of said vectors to populate one of said columns, and storing said vector,
said fitter further configured for iteratively repeating said applying and thereafter selecting and storing, until occurrence of one of
population of all said columns, and
reduction of said set to a total number of said vectors less than a total number of said columns that are currently unpopulated.
11. The apparatus of claim 10 , wherein said filter is configured to select said one vector randomly.
12. The apparatus of claim 10 , wherein said input provides an initial set that contains all possible vectors from which said columns can be populated.
13. The apparatus of claim 12 , wherein said filter operation eliminates from said initial set every vector that is a linear combination that includes any column of the identity matrix.
14. The apparatus of claim 13 , wherein said filter is configured for during an nth iteration of said applying and thereafter selecting and storing, eliminating from the associated set every vector that is a linear combination that includes one of
the vector selected during an immediately preceding iteration of said applying and thereafter selecting,
any vector selected during any other preceding iteration of said applying and thereafter selecting, and
any column of the identity matrix.
15. The apparatus of claim 14 , wherein said filter is configured to select said one vector randomly.
16. The apparatus of claim 10 , wherein said filter is configured for, during an nth iteration of said applying and thereafter selecting and storing, eliminating from the associated set every vector that is a linear combination that includes one of
the vector selected during an immediately preceding iteration of said applying and thereafter selecting,
any vector selected during any other preceding iteration of said applying and thereafter selecting, and
any column of the identity matrix.
17. The apparatus of claim 16 , wherein said filter is configured to select said one vector randomly.
18. The apparatus of claim 10 , including processing logic coupled to said filter and configured for, after all of said columns are populated, processing at least one of said stored vectors to increase a number of zeros therein, and thereafter selecting less than all of said stored vectors for use in the check matrix, wherein all of said stored vectors that are selected respectively contain a number of zeros that is less than or equal to a number of zeros contained in any of the remainder of said stored vectors.