Best Known (57−18, 57, s)-Nets in Base 128
(57−18, 57, 233019)-Net over F128 — Constructive and digital
Digital (39, 57, 233019)-net over F128, using
- 1281 times duplication [i] based on digital (38, 56, 233019)-net over F128, using
- net defined by OOA [i] based on linear OOA(12856, 233019, F128, 18, 18) (dual of [(233019, 18), 4194286, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12856, 2097171, F128, 18) (dual of [2097171, 2097115, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- 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(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(17) ⊂ Ce(12) [i] based on
- OA 9-folding and stacking [i] based on linear OA(12856, 2097171, F128, 18) (dual of [2097171, 2097115, 19]-code), using
- net defined by OOA [i] based on linear OOA(12856, 233019, F128, 18, 18) (dual of [(233019, 18), 4194286, 19]-NRT-code), using
(57−18, 57, 1270479)-Net over F128 — Digital
Digital (39, 57, 1270479)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12857, 1270479, F128, 18) (dual of [1270479, 1270422, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12857, 2097175, F128, 18) (dual of [2097175, 2097118, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- 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(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(17) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(12857, 2097175, F128, 18) (dual of [2097175, 2097118, 19]-code), using
(57−18, 57, large)-Net in Base 128 — Upper bound on s
There is no (39, 57, large)-net in base 128, because
- 16 times m-reduction [i] would yield (39, 41, large)-net in base 128, but