The (9,4)-code in Example 14.15 will, by Corollary 14.7, detect single, double, and triple errors. Hence it will correct any single error. It will not detect all errors involving four digits or correct all double errors, because 000000000 and 100001110 are two code words of Hamming distance 4 apart. For example, if the received word is 100001000, whose syndrome is 00101, Table 14.12 would decode this as 100001110 rather than 000000000; both these code words differ from the received word by a double error.

Example 14.16. Decode 100110010, 100100101, 111101100, and 000111110 using the parity check matrix in Example 14.11 and the coset leaders in Table 14.12.

BCH CODES

The most powerful class of error-correcting codes known to date were discovered around 1960 by Hocquenghem and independently by Bose and Chaudhuri. For any positive integers m and t, with t < 2m−1, there exists a Bose–Chaudhuri–Hocquenghem (BCH) code of length n = 2m − 1 that will correct any combination of t or fewer errors. These codes are polynomial codes with a generator p(x) of degree mt and have message length at least n − mt.

A t-error-correcting BCH code of length n = 2m − 1 has a generator polynomial p(x) that is constructed as follows. Take a primitive element α in the Galois field GF(2m). Let pi (x) Z2[x] be the irreducible polynomial with αi as a root, and define

p(x) = lcm(p1(x), p2(x), . . . , p2t (x)).

It is clear that α, α2, α3, . . . , α2t are all roots of p(x). By Exercise 11.56, [pi (x)]2 = pi (x2) and hence α2i is a root of pi (x). Therefore,

p(x) = lcm(p1(x), p3(x), . . . , p2t −1(x)).

Since GF(2m) is a vector space of degree m over Z2, for any β = αi , the elements 1, β, β2, . . . , βm are linearly dependent. Hence β satisfies a polynomial of degree at most m in Z2[x], and the irreducible polynomial pi (x) must also

have degree at most m. Therefore,

deg p(x) deg p1(x) · deg p3(x) · · · deg p2t −1(x) mt.

Example 14.17. Find the generator polynomials of the t-error-correcting BCH codes of length n = 15 for each value of t less than 8.

Solution. Let α be a primitive element of GF (16), where α4 + α + 1 = 0. We repeatedly refer back to the elements of GF (16) given in Table 11.4 when performing arithmetic operations in GF(16) = Z2(α).

We first calculate the irreducible polynomials pi (x) that have αi as roots. We only need to look at the odd powers of α. The element α itself is the root of

Hence

p3(x) = (x − α3)(x − α6)(x − α12)(x − α9)

=(x2 + (α3 + α6)x + α9)(x2 + (α12 + α9)x + α21)

=(x2 + α2x + α9)(x2 + α8x + α6)

=x4 + (α2 + α8)x3 + (α9 + α10 + α6)x2 + (α17 + α8)x + α15

=x4 + x3 + x2 + x + 1.

The polynomial p5(x) has roots α5, α10, and α20 = α5. Hence

p5(x) = (x − α5)(x − α10)

= x2 + x + 1.

The polynomial p7(x) has roots α7, α14, α28 = α13, α26 = α11, and α22 = α7. Hence

p7(x) = (x − α7)(x − α14)(x − α13)(x − α11)

=(x2 + αx + α6)(x2 + α4x + α9)

=x4 + x3 + 1.

Now every power of α is a root of one of the polynomials p1(x), p3(x), p5(x), or p7(x). For example, p9(x) contains α9 as a root, and therefore, p9(x) = p3(x).

The BCH code that corrects one error is generated by p(x) = p1(x) = x4 + x + 1.

The BCH code that corrects two errors is generated by

p(x) = lcm(p1(x), p3(x)) = (x4 + x + 1)(x4 + x3 + x2 + x + 1).

This least common multiple is the product because p1(x) and p3(x) are different irreducible polynomials. Hence p(x) = x8 + x7 + x6 + x4 + 1.

The BCH code that corrects three errors is generated by

p(x) = lcm(p1(x), p3(x), p5(x))

=(x4 + x + 1)(x4 + x3 + x2 + x + 1)(x2 + x + 1)

=x10 + x8 + x5 + x4 + x2 + x + 1.

The BCH code that corrects four errors is generated by

p(x) = lcm(p1(x), p3(x), p5(x), p7(x))

= p1(x) · p3(x) · p5(x) · p7(x)

This polynomial contains all the elements of GF (16) as roots, except for 0 and 1. Since p9(x) = p3(x), the five-error-correcting BCH code is generated by

p(x) = lcm(p1(x), p3(x), p5(x), p7(x), p9(x))

= (x15 + 1)/(x + 1),

and this is also the generator of the sixand seven-error-correcting BCH codes. These results are summarized in Table 14.14.

For example, the two-error-correcting BCH code is a (15, 7)-code with generator polynomial x8 + x7 + x6 + x4 + 1. It contains seven message digits and eight check digits.

The seven-error-correcting code generated by (x15 + 1)/(x + 1) has message length 1, and the two code words are the sequence of 15 zeros and the sequence of 15 ones. Each received word can be decoded by majority rule to give the

TABLE 14.14. Construction of t-Error-Correcting BCH Codes of Length 15