Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes

We present
O
(
m
³
) algorithms for specifying the support of minimum-weight codewords of extended binary BCH codes of length n = 2m and designed distance
d
(
m
,
s
,
i
) := 2
^m−1−s
−2
^{m−1−i−s}
for some values of
m
,
i
,
s
, where
m
may grow to infinity. Here, the support is specified as the sum of two sets: a set of 2
²ⁱ⁻¹
−2
ⁱ⁻¹
elements, and a subspace of dimension
m
− 2
i
−
s
, specified by a basis. In some detail, for designed distance 6 · 2
^j
,
j
∈ {0, . . . ,
m
− 4}, we have a deterministic algorithm for even
m
≥ 4, and a probabilistic algorithm with success probability 1 − O(2
^−m
) for odd
m
> 4. For designed distance 28 · 2
^j
,
j
∈ {0, . . . ,
m
−6}, we have a probabilistic algorithm with success probability ≥ 1/3 − O(2
^−m/2
) for even
m
≥ 6. Finally, for designed distance 120 · 2
^j
,
j
∈ {0, . . . ,
m
− 8}, we have a deterministic algorithm for
m
≥ 8 divisible by 4. We also show how Gold functions can be used to find the support of minimum-weight words for designed distance
d
(
m
,
s
,
i
) (for
i
∈ {0, . . . , ⌊
m
/2⌋}, and
s
≤
m
− 2
i
) whenever 2
i
|
m
. Our construction builds on results of Kasami and Lin (IEEE Trans. IT, 1972), who proved that for extended binary BCH codes of designed distance
d
(
m
,
s
,
i
) (for integers
m
≥ 2, 0 ≤
i
≤ ⌊
m
/2⌋, and 0 ≤
s
≤
m
− 2
i
), the minimum distance equals the designed distance. The proof of Kasami and Lin makes use of a non-constructive existence result of Berlekamp (Infom. Contrl., 1970), and a constructive “down-conversion theorem” that converts some words in BCH codes to lower-weight words in BCH codes of lower designed distance. Our main contribution is in replacing the non-constructive counting argument of Berlekamp by a low-complexity algorithm. In one aspect, the current paper extends the results of Grigorescu and Kaufman (IEEE Trans. IT, 2012), who presented explicit minimum-weight codewords for extended binary BCH codes of designed distance
exactly
6 (and hence also for designed distance 6 · 2
^j
, by a well-known “up-conversion theorem”), as we cover more cases of the minimum distance. In fact, we prove that the codeword constructed by Grigorescu and Kaufman is a special case of the current construction. However, the minimum-weight codewords we construct do not generate the code, and are not affine generators, except, possibly, for a designed distance of 6.