Best Known (27, 27+13, s)-Nets in Base 128
(27, 27+13, 349527)-Net over F128 — Constructive and digital
Digital (27, 40, 349527)-net over F128, using
- 1281 times duplication [i] based on digital (26, 39, 349527)-net over F128, using
- net defined by OOA [i] based on linear OOA(12839, 349527, F128, 13, 13) (dual of [(349527, 13), 4543812, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12839, 2097163, F128, 13) (dual of [2097163, 2097124, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(12839, 2097163, F128, 13) (dual of [2097163, 2097124, 14]-code), using
- net defined by OOA [i] based on linear OOA(12839, 349527, F128, 13, 13) (dual of [(349527, 13), 4543812, 14]-NRT-code), using
(27, 27+13, 1143473)-Net over F128 — Digital
Digital (27, 40, 1143473)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12840, 1143473, F128, 13) (dual of [1143473, 1143433, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12840, 2097168, F128, 13) (dual of [2097168, 2097128, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12840, 2097168, F128, 13) (dual of [2097168, 2097128, 14]-code), using
(27, 27+13, large)-Net in Base 128 — Upper bound on s
There is no (27, 40, large)-net in base 128, because
- 11 times m-reduction [i] would yield (27, 29, large)-net in base 128, but