Correction of errors of weight 2

3.3 Correction of errors of weight 2

A binary extended BCH code over GF(2) with minimum distance 5 can be decoded in a simple way by factoring a quadratic equation over GF(2). An extended Preparata code over Z₄can be decoded in a similar simple way by dropping combinations of the Z₄syndromes into GF(2), then factoring a quadratic equation over GF(2). We shall develop these decoding procedures for distance-5 codes both over GF(2) and over Z4

in this section. In each case, because the extended code must have even weight, the existence of a two-error-correcting decoder implies further that the extended code has weight 6 so it can detect triple errors. This means that the decoders that we describe

149 3.3 Correction of errors of weight 2

can be easily extended – though we will not do so – to detect triple errors as well as correct double errors.

A BCH code of minimum distance 5 over GF(2) is the set of polynomials c(x) such that c(α) = c(α³) = 0. The syndromes are S1=v(α) = e(α) and S3=v(α³) = e(α³).

For a single-error pattern, e(x) = xⁱ, so S₁ = αⁱ and S₃ = α³ⁱ. This case can be recognized by noting that S₁³+ S3 = 0. Then the exponent of α, in S1, points to the location of the error. For a double-error pattern, e(x) = xⁱ + x^i. The syndromes are S₁ = X1+ X2, which is nonzero, and S₃ = X₁³+ X₂³, where X₁ = αⁱ and X₂ = α^i. This case can be recognized by noting that S₁³+ S3= 0. Then

(x − X1)(x − X2) = x²+ S1x+ (S₁³+ S3)/S1.

The polynomial on the right side depends only on the syndromes. By factoring the right side, as by trial and error, one obtains X₁ and X₂. The exponents ofα in X1 and X₂ point to the two error locations. Thus this process forms a decoding algorithm for a distance-5 BCH code.

We will now give a comparable decoding algorithm for an extended Preparata code over the ring Z₄ that corrects all error patterns of Lee weight 2 or less. This means that the minimum Lee distance of the code is at least 5, and since the min-imum distance of the extended code must be even, it is at least 6. This decoding algorithm provides an alternative proof that the minimum distance of the extended Preparata code is (at least) 6. The decoding algorithm also applies to an extended Preparata code over GF(2^m) simply by regarding each pair of bits as a symbol of Z₄.

Letv(x) be any senseword polynomial with Lee distance at most 2 from a code-word c(x) of the Preparata code. The senseword polynomialv(x) can be regarded as a codeword polynomial c(x) corrupted by an error polynomial asv(x) = c(x) + e(x).

The correctible sensewords consist of two cases: single errors, either with e(x) = 2xⁱ or with e(x) = ±xⁱ; and double errors, with e(x) = ±xⁱ ± x^i. (Detectable error patterns, which we do not consider here, have Lee weight 3 and consist of double errors with e(x) = 2xⁱ± x^i and triple errors with e(x) = ±xⁱ± x^i ± x^i.) Because the codewords are defined by c_∞ + c(1) = 0 and c(ξ) = c(ξ²) = 0, the sense-word v(x) can be reduced to three pieces of data. These are the three syndromes S₀ = v_∞+v(1) = e_∞+ e(1), S1 = v(ξ) = e(ξ), and S2 = v(ξ²) = e(ξ²). We will give an algorithm to compute e(x) from S0, S₁, and S₂. (Although syndrome S₂ contains no information not already in S₁, it is more convenient to use both S₂and S₁ as the input to the decoder.)

The syndrome S₀ is an element of Z₄, corresponding to one of three situations. If S₀ = 0, either there are no errors or there are two errors with values +1 and −1. If S₀ = ±1, then there is one error with value ±1. If S0 = 2, then either there are two errors, both with the same value+1 or −1, or there is a single error with value 2.

Suppose there is a single error, so e(x) = eixⁱ, and S₀= ei,

S₁= e(ξ) = eiξⁱ, S₂= e(ξ²) = eiξ²ⁱ,

where e_i = ±1 or 2. The case of a single error of Lee weight 1 can be recognized by noting that S₀is ±1, or by computing S₁^2m and finding that S₁^2m = S1. The case of a single error of Lee weight 2 can be recognized by noting that S₀ = 2 and S₁² = 0.

If either instance of a single error is observed, thenξⁱ = S1/S0. The exponent ofξ uniquely points to the single term of e(x) with a nonzero coefficient. Syndrome S0is the value of the error.

For the case of a correctable pattern with two errors, S₀ = 0 or 2 and S₁^2m = S1. Because the Lee weight is 2, e_i = ±1 and ei^= ±1. Then

S₀= e(1) = ±1 ± 1, S₁= e(ξ) = ±ξⁱ± ξ^i, S₂= e(ξ²) = ±ξ²ⁱ± ξ^2i.

Syndrome S₀ = 0 if the two errors have opposite signs. In this case, without loss of generality, we write e(x) = xⁱ − x^i. If, instead, the two errors have the same sign, syndrome S₂= 2. Thus for any correctable pattern with two errors, we have

S₁²− S2 = ξ²ⁱ+ 2ξⁱξ^i+ ξ^2i∓ ξ²ⁱ∓ ξ^2i

=

⎧⎪

⎨

⎪⎩

2(ξⁱξ^i+ ξ^2i) if S₀= 0

2(ξⁱξ^i) if S₀= 2 and both errors are + 1 2(ξ²ⁱ+ ξⁱξ^i+ ξ^2i) if S0= 2 and both errors are − 1.

In any case, this is an element of Z₄that has only an even part, which is twice the odd part of the term in parentheses. Every even element of GR(4^m) is a unique element of GF(2^m) multiplied by 2, so the terms in parentheses can be dropped into GF(2^m). Let B be the term S₁²− S2, reduced by the factor of 2. Then

B=

⎧⎪

⎨

⎪⎩

αⁱα^i+ α²ⁱ if S₀= 0

αⁱα^i if S₀= 2 and both errors are + 1 α²ⁱ+ αⁱα^i+ α^2i if S₀= 2 and both errors are − 1.

Let A denote S₁modulo 2 and let X₁and X₂ denoteξⁱ andξ^i modulo 2. Then the equations, now in GF(2), become

A= X1+ X2

151 3.4 The Sugiyama algorithm

and

B+ X₁²= X1X₂ if S₀= 0,

B= X1X₂ if S₀= 2 and both errors are + 1, B+ A²= X1X₂ if S₀= 2 and both errors are − 1.

It is trivial to solve the first case by substituting X₂ = A + X1into B+ X₁² = X1X₂. This yields X₁ = B/A and X2 = A + B/A. The latter two cases, those with S0 = 2, though not distinguished by the value of S₀, can be distinguished because only one of them has a solution. To see this, reduce the equations to observe that, in the two cases, X₁and X₂are the solutions of either

z²+ Az + B = 0 or

z²+ Az + A²+ B = 0.

The first of these can be solved only if trace(B/A²) = 0. The second can be solved only if trace(B/A²) = 1. Only one of these can be true, so only one of the two polynomials has its zeros in the locator field.