Best Known (46−23, 46, s)-Nets in Base 128
(46−23, 46, 1489)-Net over F128 — Constructive and digital
Digital (23, 46, 1489)-net over F128, using
- 1281 times duplication [i] based on digital (22, 45, 1489)-net over F128, using
- net defined by OOA [i] based on linear OOA(12845, 1489, F128, 23, 23) (dual of [(1489, 23), 34202, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12845, 16380, F128, 23) (dual of [16380, 16335, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12845, 16380, F128, 23) (dual of [16380, 16335, 24]-code), using
- net defined by OOA [i] based on linear OOA(12845, 1489, F128, 23, 23) (dual of [(1489, 23), 34202, 24]-NRT-code), using
(46−23, 46, 4097)-Net over F128 — Digital
Digital (23, 46, 4097)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12846, 4097, F128, 4, 23) (dual of [(4097, 4), 16342, 24]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12846, 16388, F128, 23) (dual of [16388, 16342, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12846, 16390, F128, 23) (dual of [16390, 16344, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(12845, 16385, F128, 23) (dual of [16385, 16340, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(12841, 16385, F128, 21) (dual of [16385, 16344, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12846, 16390, F128, 23) (dual of [16390, 16344, 24]-code), using
- OOA 4-folding [i] based on linear OA(12846, 16388, F128, 23) (dual of [16388, 16342, 24]-code), using
(46−23, 46, large)-Net in Base 128 — Upper bound on s
There is no (23, 46, large)-net in base 128, because
- 21 times m-reduction [i] would yield (23, 25, large)-net in base 128, but