Best Known (75−23, 75, s)-Nets in Base 128
(75−23, 75, 190653)-Net over F128 — Constructive and digital
Digital (52, 75, 190653)-net over F128, using
- 1281 times duplication [i] based on digital (51, 74, 190653)-net over F128, using
- net defined by OOA [i] based on linear OOA(12874, 190653, F128, 23, 23) (dual of [(190653, 23), 4384945, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12874, 2097184, F128, 23) (dual of [2097184, 2097110, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,7]) [i] based on
- 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(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,11]) ⊂ C([0,7]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(12874, 2097184, F128, 23) (dual of [2097184, 2097110, 24]-code), using
- net defined by OOA [i] based on linear OOA(12874, 190653, F128, 23, 23) (dual of [(190653, 23), 4384945, 24]-NRT-code), using
(75−23, 75, 1820091)-Net over F128 — Digital
Digital (52, 75, 1820091)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12875, 1820091, F128, 23) (dual of [1820091, 1820016, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12875, 2097187, F128, 23) (dual of [2097187, 2097112, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(13) [i] based on
- linear OA(12867, 2097152, F128, 23) (dual of [2097152, 2097085, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(1288, 35, F128, 8) (dual of [35, 27, 9]-code or 35-arc in PG(7,128)), using
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- Reed–Solomon code RS(120,128) [i]
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- construction X applied to Ce(22) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(12875, 2097187, F128, 23) (dual of [2097187, 2097112, 24]-code), using
(75−23, 75, large)-Net in Base 128 — Upper bound on s
There is no (52, 75, large)-net in base 128, because
- 21 times m-reduction [i] would yield (52, 54, large)-net in base 128, but