Best Known (54−27, 54, s)-Nets in Base 256
(54−27, 54, 5041)-Net over F256 — Constructive and digital
Digital (27, 54, 5041)-net over F256, using
- 2561 times duplication [i] based on digital (26, 53, 5041)-net over F256, using
- net defined by OOA [i] based on linear OOA(25653, 5041, F256, 27, 27) (dual of [(5041, 27), 136054, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25653, 65534, F256, 27) (dual of [65534, 65481, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25653, 65536, F256, 27) (dual of [65536, 65483, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(25653, 65536, F256, 27) (dual of [65536, 65483, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25653, 65534, F256, 27) (dual of [65534, 65481, 28]-code), using
- net defined by OOA [i] based on linear OOA(25653, 5041, F256, 27, 27) (dual of [(5041, 27), 136054, 28]-NRT-code), using
(54−27, 54, 13108)-Net over F256 — Digital
Digital (27, 54, 13108)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25654, 13108, F256, 5, 27) (dual of [(13108, 5), 65486, 28]-NRT-code), using
- OOA 5-folding [i] based on linear OA(25654, 65540, F256, 27) (dual of [65540, 65486, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25654, 65542, F256, 27) (dual of [65542, 65488, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25654, 65542, F256, 27) (dual of [65542, 65488, 28]-code), using
- OOA 5-folding [i] based on linear OA(25654, 65540, F256, 27) (dual of [65540, 65486, 28]-code), using
(54−27, 54, large)-Net in Base 256 — Upper bound on s
There is no (27, 54, large)-net in base 256, because
- 25 times m-reduction [i] would yield (27, 29, large)-net in base 256, but