Best Known (62−31, 62, s)-Nets in Base 128
(62−31, 62, 1092)-Net over F128 — Constructive and digital
Digital (31, 62, 1092)-net over F128, using
- 1281 times duplication [i] based on digital (30, 61, 1092)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 1092, F128, 31, 31) (dual of [(1092, 31), 33791, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12861, 16381, F128, 31) (dual of [16381, 16320, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12861, 16381, F128, 31) (dual of [16381, 16320, 32]-code), using
- net defined by OOA [i] based on linear OOA(12861, 1092, F128, 31, 31) (dual of [(1092, 31), 33791, 32]-NRT-code), using
(62−31, 62, 4097)-Net over F128 — Digital
Digital (31, 62, 4097)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12862, 4097, F128, 4, 31) (dual of [(4097, 4), 16326, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12862, 16388, F128, 31) (dual of [16388, 16326, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(12862, 16390, F128, 31) (dual of [16390, 16328, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- linear OA(12861, 16385, F128, 31) (dual of [16385, 16324, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(12857, 16385, F128, 29) (dual of [16385, 16328, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 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 C([0,15]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12862, 16390, F128, 31) (dual of [16390, 16328, 32]-code), using
- OOA 4-folding [i] based on linear OA(12862, 16388, F128, 31) (dual of [16388, 16326, 32]-code), using
(62−31, 62, large)-Net in Base 128 — Upper bound on s
There is no (31, 62, large)-net in base 128, because
- 29 times m-reduction [i] would yield (31, 33, large)-net in base 128, but