Best Known (28, 42, s)-Nets in Base 128
(28, 42, 299594)-Net over F128 — Constructive and digital
Digital (28, 42, 299594)-net over F128, using
- 1281 times duplication [i] based on digital (27, 41, 299594)-net over F128, using
- net defined by OOA [i] based on linear OOA(12841, 299594, F128, 14, 14) (dual of [(299594, 14), 4194275, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(12841, 2097158, F128, 14) (dual of [2097158, 2097117, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(12841, 2097158, F128, 14) (dual of [2097158, 2097117, 15]-code), using
- net defined by OOA [i] based on linear OOA(12841, 299594, F128, 14, 14) (dual of [(299594, 14), 4194275, 15]-NRT-code), using
(28, 42, 1048581)-Net over F128 — Digital
Digital (28, 42, 1048581)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12842, 1048581, F128, 2, 14) (dual of [(1048581, 2), 2097120, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12842, 2097162, F128, 14) (dual of [2097162, 2097120, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12842, 2097163, F128, 14) (dual of [2097163, 2097121, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(12842, 2097163, F128, 14) (dual of [2097163, 2097121, 15]-code), using
- OOA 2-folding [i] based on linear OA(12842, 2097162, F128, 14) (dual of [2097162, 2097120, 15]-code), using
(28, 42, large)-Net in Base 128 — Upper bound on s
There is no (28, 42, large)-net in base 128, because
- 12 times m-reduction [i] would yield (28, 30, large)-net in base 128, but