Best Known (50, 75, s)-Nets in Base 128
(50, 75, 174763)-Net over F128 — Constructive and digital
Digital (50, 75, 174763)-net over F128, using
- 1281 times duplication [i] based on digital (49, 74, 174763)-net over F128, using
- net defined by OOA [i] based on linear OOA(12874, 174763, F128, 25, 25) (dual of [(174763, 25), 4369001, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12874, 2097157, F128, 25) (dual of [2097157, 2097083, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12874, 2097160, F128, 25) (dual of [2097160, 2097086, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(12873, 2097153, F128, 25) (dual of [2097153, 2097080, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12867, 2097153, F128, 23) (dual of [2097153, 2097086, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [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 C([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12874, 2097160, F128, 25) (dual of [2097160, 2097086, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12874, 2097157, F128, 25) (dual of [2097157, 2097083, 26]-code), using
- net defined by OOA [i] based on linear OOA(12874, 174763, F128, 25, 25) (dual of [(174763, 25), 4369001, 26]-NRT-code), using
(50, 75, 700082)-Net over F128 — Digital
Digital (50, 75, 700082)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12875, 700082, F128, 2, 25) (dual of [(700082, 2), 1400089, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12875, 1048581, F128, 2, 25) (dual of [(1048581, 2), 2097087, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12875, 2097162, F128, 25) (dual of [2097162, 2097087, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12875, 2097163, F128, 25) (dual of [2097163, 2097088, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(12873, 2097152, F128, 25) (dual of [2097152, 2097079, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(12875, 2097163, F128, 25) (dual of [2097163, 2097088, 26]-code), using
- OOA 2-folding [i] based on linear OA(12875, 2097162, F128, 25) (dual of [2097162, 2097087, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(12875, 1048581, F128, 2, 25) (dual of [(1048581, 2), 2097087, 26]-NRT-code), using
(50, 75, large)-Net in Base 128 — Upper bound on s
There is no (50, 75, large)-net in base 128, because
- 23 times m-reduction [i] would yield (50, 52, large)-net in base 128, but