Coding Theory by John C. Bowman

By John C. Bowman

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To find the inverse of 5 in Z26 we use the Euclidean division algorithm: 1 = 5x + 26y, 26 = 5 · 5 + 1 ⇒ 1 = 26 − 5 · 5, from which we see that x = −5 is a solution. Thus e−1 = −5 4 −1 6 5 = . −3 2 15 16 We can use e to encrypt the word “SECRET”, in other words the message 18 4 2 17 4 19, by breaking the message up into vectors of length two: [18 4], [2 17], [4 19] and then multiplying the transpose of each vector by e on the left. The result is [14 18], [21 22], [1 10], or the cipher text “OSVWBK”.

Because Reed–Solomon codes are optimal in this sense and easily implementable in hardware (using shift registers), they are widely used for error correction in computer memory chips, magnetic and optical (compact disk) storage systems, high-speed modems, and data transmission channels. • Since 2 is a primitive element of Z11 , the polynomial g(x) = (x−2)(x−4)(x−8)(x−5)(x−10)(x−9) = x6 +6x5 +5x4 +7x3 +2x2 +8x+2 generates a triple-error correcting [10, 4, 7] Reed–Solomon code over Z11 . One of the codewords is g(x) itself, which has weight 7, so the design distance in this case is the actual minimum distance.

In fact, since the second row of G24 is a codeword of weight 8, we see that the minimum distance of C24 is exactly 8. Exercise: Show that the ternary Golay [11, 6] code generated by the first 11 columns of the generator matrix   1 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 0 1 0 1 2 2 1   0 0 1 0 0 0 1 1 0 1 2 2  G12 =  0 0 0 1 0 0 1 2 1 0 1 2   0 0 0 0 1 0 1 2 2 1 0 1 0 0 0 0 0 1 1 1 2 2 1 0 has minimum distance 5. 2 (Nonexistence of (90, 278 , 5) codes) There exist no (90, 278 , 5) codes.

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