Best Known (51, 51+24, s)-Nets in Base 128
(51, 51+24, 174764)-Net over F128 — Constructive and digital
Digital (51, 75, 174764)-net over F128, using
- 1281 times duplication [i] based on digital (50, 74, 174764)-net over F128, using
- net defined by OOA [i] based on linear OOA(12874, 174764, F128, 24, 24) (dual of [(174764, 24), 4194262, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(12874, 2097168, F128, 24) (dual of [2097168, 2097094, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12874, 2097171, F128, 24) (dual of [2097171, 2097097, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(12874, 2097171, F128, 24) (dual of [2097171, 2097097, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(12874, 2097168, F128, 24) (dual of [2097168, 2097094, 25]-code), using
- net defined by OOA [i] based on linear OOA(12874, 174764, F128, 24, 24) (dual of [(174764, 24), 4194262, 25]-NRT-code), using
(51, 51+24, 1048587)-Net over F128 — Digital
Digital (51, 75, 1048587)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12875, 1048587, F128, 2, 24) (dual of [(1048587, 2), 2097099, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12875, 2097174, F128, 24) (dual of [2097174, 2097099, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12875, 2097175, F128, 24) (dual of [2097175, 2097100, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(17) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to Ce(23) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(12875, 2097175, F128, 24) (dual of [2097175, 2097100, 25]-code), using
- OOA 2-folding [i] based on linear OA(12875, 2097174, F128, 24) (dual of [2097174, 2097099, 25]-code), using
(51, 51+24, large)-Net in Base 128 — Upper bound on s
There is no (51, 75, large)-net in base 128, because
- 22 times m-reduction [i] would yield (51, 53, large)-net in base 128, but