Best Known (56−25, 56, s)-Nets in Base 128
(56−25, 56, 1367)-Net over F128 — Constructive and digital
Digital (31, 56, 1367)-net over F128, using
- net defined by OOA [i] based on linear OOA(12856, 1367, F128, 25, 25) (dual of [(1367, 25), 34119, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12856, 16405, F128, 25) (dual of [16405, 16349, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12856, 16408, F128, 25) (dual of [16408, 16352, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(12849, 16385, F128, 25) (dual of [16385, 16336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(1287, 23, F128, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12856, 16408, F128, 25) (dual of [16408, 16352, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12856, 16405, F128, 25) (dual of [16405, 16349, 26]-code), using
(56−25, 56, 5461)-Net in Base 128 — Constructive
(31, 56, 5461)-net in base 128, using
- base change [i] based on digital (24, 49, 5461)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 5461, F256, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25649, 65533, F256, 25) (dual of [65533, 65484, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25649, 65533, F256, 25) (dual of [65533, 65484, 26]-code), using
- net defined by OOA [i] based on linear OOA(25649, 5461, F256, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
(56−25, 56, 8204)-Net over F128 — Digital
Digital (31, 56, 8204)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12856, 8204, F128, 2, 25) (dual of [(8204, 2), 16352, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12856, 16408, F128, 25) (dual of [16408, 16352, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(12849, 16385, F128, 25) (dual of [16385, 16336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(1287, 23, F128, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- OOA 2-folding [i] based on linear OA(12856, 16408, F128, 25) (dual of [16408, 16352, 26]-code), using
(56−25, 56, 11731)-Net in Base 128
(31, 56, 11731)-net in base 128, using
- base change [i] based on digital (24, 49, 11731)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25649, 11731, F256, 5, 25) (dual of [(11731, 5), 58606, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25649, 13107, F256, 5, 25) (dual of [(13107, 5), 65486, 26]-NRT-code), using
- OOA 5-folding [i] based on linear OA(25649, 65535, F256, 25) (dual of [65535, 65486, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using
- OOA 5-folding [i] based on linear OA(25649, 65535, F256, 25) (dual of [65535, 65486, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(25649, 13107, F256, 5, 25) (dual of [(13107, 5), 65486, 26]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25649, 11731, F256, 5, 25) (dual of [(11731, 5), 58606, 26]-NRT-code), using
(56−25, 56, large)-Net in Base 128 — Upper bound on s
There is no (31, 56, large)-net in base 128, because
- 23 times m-reduction [i] would yield (31, 33, large)-net in base 128, but