Best Known (39, 39+37, s)-Nets in Base 128
(39, 39+37, 910)-Net over F128 — Constructive and digital
Digital (39, 76, 910)-net over F128, using
- 1283 times duplication [i] based on digital (36, 73, 910)-net over F128, using
- net defined by OOA [i] based on linear OOA(12873, 910, F128, 37, 37) (dual of [(910, 37), 33597, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(12873, 16381, F128, 37) (dual of [16381, 16308, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(12873, 16384, F128, 37) (dual of [16384, 16311, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- discarding factors / shortening the dual code based on linear OA(12873, 16384, F128, 37) (dual of [16384, 16311, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(12873, 16381, F128, 37) (dual of [16381, 16308, 38]-code), using
- net defined by OOA [i] based on linear OOA(12873, 910, F128, 37, 37) (dual of [(910, 37), 33597, 38]-NRT-code), using
(39, 39+37, 4737)-Net over F128 — Digital
Digital (39, 76, 4737)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12876, 4737, F128, 3, 37) (dual of [(4737, 3), 14135, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12876, 5465, F128, 3, 37) (dual of [(5465, 3), 16319, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12876, 16395, F128, 37) (dual of [16395, 16319, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(12876, 16396, F128, 37) (dual of [16396, 16320, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(12873, 16385, F128, 37) (dual of [16385, 16312, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (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([0,18]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12876, 16396, F128, 37) (dual of [16396, 16320, 38]-code), using
- OOA 3-folding [i] based on linear OA(12876, 16395, F128, 37) (dual of [16395, 16319, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(12876, 5465, F128, 3, 37) (dual of [(5465, 3), 16319, 38]-NRT-code), using
(39, 39+37, large)-Net in Base 128 — Upper bound on s
There is no (39, 76, large)-net in base 128, because
- 35 times m-reduction [i] would yield (39, 41, large)-net in base 128, but