Best Known (56−33, 56, s)-Nets in Base 256
(56−33, 56, 521)-Net over F256 — Constructive and digital
Digital (23, 56, 521)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (3, 19, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- digital (4, 37, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- digital (3, 19, 260)-net over F256, using
(56−33, 56, 840)-Net over F256 — Digital
Digital (23, 56, 840)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25656, 840, F256, 33) (dual of [840, 784, 34]-code), using
- 64 step Varšamov–Edel lengthening with (ri) = (1, 63 times 0) [i] based on linear OA(25655, 775, F256, 33) (dual of [775, 720, 34]-code), using
- construction XX applied to C1 = C([242,273]), C2 = C([241,272]), C3 = C1 + C2 = C([242,272]), and C∩ = C1 ∩ C2 = C([241,273]) [i] based on
- linear OA(25653, 771, F256, 32) (dual of [771, 718, 33]-code), using the BCH-code C(I) with length 771 | 2562−1, defining interval I = {242,243,…,273}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(25653, 771, F256, 32) (dual of [771, 718, 33]-code), using the BCH-code C(I) with length 771 | 2562−1, defining interval I = {241,242,…,272}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(25655, 771, F256, 33) (dual of [771, 716, 34]-code), using the BCH-code C(I) with length 771 | 2562−1, defining interval I = {241,242,…,273}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(25651, 771, F256, 31) (dual of [771, 720, 32]-code), using the BCH-code C(I) with length 771 | 2562−1, defining interval I = {242,243,…,272}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([242,273]), C2 = C([241,272]), C3 = C1 + C2 = C([242,272]), and C∩ = C1 ∩ C2 = C([241,273]) [i] based on
- 64 step Varšamov–Edel lengthening with (ri) = (1, 63 times 0) [i] based on linear OA(25655, 775, F256, 33) (dual of [775, 720, 34]-code), using
(56−33, 56, 5062007)-Net in Base 256 — Upper bound on s
There is no (23, 56, 5062008)-net in base 256, because
- 1 times m-reduction [i] would yield (23, 55, 5062008)-net in base 256, but
- the generalized Rao bound for nets shows that 256m ≥ 2 839219 110778 830163 688950 479691 383538 763713 071457 258046 950529 268559 277560 282019 974560 105156 775019 740384 149043 502369 283972 142648 194391 > 25655 [i]