Best Known (32, 63, s)-Nets in Base 256
(32, 63, 4369)-Net over F256 — Constructive and digital
Digital (32, 63, 4369)-net over F256, using
- 2562 times duplication [i] based on digital (30, 61, 4369)-net over F256, using
- net defined by OOA [i] based on linear OOA(25661, 4369, F256, 31, 31) (dual of [(4369, 31), 135378, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- OOA 15-folding and stacking with additional row [i] based on linear OA(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using
- net defined by OOA [i] based on linear OOA(25661, 4369, F256, 31, 31) (dual of [(4369, 31), 135378, 32]-NRT-code), using
(32, 63, 13108)-Net over F256 — Digital
Digital (32, 63, 13108)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25663, 13108, F256, 5, 31) (dual of [(13108, 5), 65477, 32]-NRT-code), using
- 2561 times duplication [i] based on linear OOA(25662, 13108, F256, 5, 31) (dual of [(13108, 5), 65478, 32]-NRT-code), using
- OOA 5-folding [i] based on linear OA(25662, 65540, F256, 31) (dual of [65540, 65478, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(25662, 65542, F256, 31) (dual of [65542, 65480, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- linear OA(25661, 65537, F256, 31) (dual of [65537, 65476, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25662, 65542, F256, 31) (dual of [65542, 65480, 32]-code), using
- OOA 5-folding [i] based on linear OA(25662, 65540, F256, 31) (dual of [65540, 65478, 32]-code), using
- 2561 times duplication [i] based on linear OOA(25662, 13108, F256, 5, 31) (dual of [(13108, 5), 65478, 32]-NRT-code), using
(32, 63, large)-Net in Base 256 — Upper bound on s
There is no (32, 63, large)-net in base 256, because
- 29 times m-reduction [i] would yield (32, 34, large)-net in base 256, but