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By Mirza Bashir-ud-din Mahmud Ahmad

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The method of construction is best understood through an example. Consider the array v2 v 3 Vlv3 4 - 1 0 0 0o 4---1 0 0 V1V2 V3 v2 v1 vo 1 1 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1- 1 0 1 0 0 O (100) which is formed by taking v 1 , v 2 , and v 3 in such a way that the set of 3-tuples in their rows is the set of all nonzero 3-tuples, and then adding the all "one" column vector to the list and the column vectors formed by taking all vector inner products of v 1 , v 2 , and v 3 taken two at a time.

Except for e(l), e 6 (1) the only information The same process can be repeated in the second-to-last and third-to-last rows of the parity triangles. 7. Thus, in all, for this code: (3)2 32 parity checks of size 22 , plus ( 3 )z 3 -1 parity checks of size 21, plus (3)2 -1 parity checks of size 20, all orthogonal on e 1 , can be formed. When this example is generalized so that the array contains all vector inner products of vl, v 2 , ... , vM taken K or fewer at a time, we obtain Theorem 14. THEOREM 14: For any integers M and K with the property that K _ M, there exists a convolutional code with R = d= ; (rM) 1/ 2 M and nA = nE =L 2M (M) 2M so that (101) and so that the code can be completely orthogonalized.

The same process can be carried out on the second-to-last, the third-to-last, etc. rows of the parity triangles. reduce to the K-l order code. After (M) rows have been processed, the remaining rows The union of the sets of orthogonal parity checks, formed from each set of rows of the parity triangles, itself is orthogonal on e (1), since there is only one information noise bit checked by the set formed from each set of rows, and these bits are all distinct. The total number, J, of parity checks orthogonal on e (l) that are formed in this manner is K E ( ) M (102) j=0 which is one less than the sum in Eq.

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Anwarul 'Ulum Volume 5: Sadaqat-e-Islam by Mirza Bashir-ud-din Mahmud Ahmad

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