Best Known (40, 58, s)-Nets in Base 128
(40, 58, 233019)-Net over F128 — Constructive and digital
Digital (40, 58, 233019)-net over F128, using
- 1282 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
(40, 58, 1720556)-Net over F128 — Digital
Digital (40, 58, 1720556)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12858, 1720556, F128, 18) (dual of [1720556, 1720498, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12858, 2097179, F128, 18) (dual of [2097179, 2097121, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [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(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1286, 27, F128, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,128)), using
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- Reed–Solomon code RS(122,128) [i]
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(12858, 2097179, F128, 18) (dual of [2097179, 2097121, 19]-code), using
(40, 58, large)-Net in Base 128 — Upper bound on s
There is no (40, 58, large)-net in base 128, because
- 16 times m-reduction [i] would yield (40, 42, large)-net in base 128, but