Best Known (20, 20+17, s)-Nets in Base 128
(20, 20+17, 2049)-Net over F128 — Constructive and digital
Digital (20, 37, 2049)-net over F128, using
- 1281 times duplication [i] based on digital (19, 36, 2049)-net over F128, using
- net defined by OOA [i] based on linear OOA(12836, 2049, F128, 17, 17) (dual of [(2049, 17), 34797, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12836, 16393, F128, 17) (dual of [16393, 16357, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(12836, 16396, F128, 17) (dual of [16396, 16360, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-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([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12836, 16396, F128, 17) (dual of [16396, 16360, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12836, 16393, F128, 17) (dual of [16393, 16357, 18]-code), using
- net defined by OOA [i] based on linear OOA(12836, 2049, F128, 17, 17) (dual of [(2049, 17), 34797, 18]-NRT-code), using
(20, 20+17, 8199)-Net over F128 — Digital
Digital (20, 37, 8199)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12837, 8199, F128, 2, 17) (dual of [(8199, 2), 16361, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12837, 16398, F128, 17) (dual of [16398, 16361, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(12833, 16384, F128, 17) (dual of [16384, 16351, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1284, 14, F128, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(12837, 16398, F128, 17) (dual of [16398, 16361, 18]-code), using
(20, 20+17, large)-Net in Base 128 — Upper bound on s
There is no (20, 37, large)-net in base 128, because
- 15 times m-reduction [i] would yield (20, 22, large)-net in base 128, but