Best Known (40, 40+35, s)-Nets in Base 128
(40, 40+35, 964)-Net over F128 — Constructive and digital
Digital (40, 75, 964)-net over F128, using
- 1285 times duplication [i] based on digital (35, 70, 964)-net over F128, using
- net defined by OOA [i] based on linear OOA(12870, 964, F128, 35, 35) (dual of [(964, 35), 33670, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(12870, 16389, F128, 35) (dual of [16389, 16319, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(12870, 16390, F128, 35) (dual of [16390, 16320, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- linear OA(12869, 16385, F128, 35) (dual of [16385, 16316, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (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(1281, 5, F128, 1) (dual of [5, 4, 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([0,17]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12870, 16390, F128, 35) (dual of [16390, 16320, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(12870, 16389, F128, 35) (dual of [16389, 16319, 36]-code), using
- net defined by OOA [i] based on linear OOA(12870, 964, F128, 35, 35) (dual of [(964, 35), 33670, 36]-NRT-code), using
(40, 40+35, 6444)-Net over F128 — Digital
Digital (40, 75, 6444)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12875, 6444, F128, 2, 35) (dual of [(6444, 2), 12813, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12875, 8202, F128, 2, 35) (dual of [(8202, 2), 16329, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12875, 16404, F128, 35) (dual of [16404, 16329, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(27) [i] based on
- linear OA(12869, 16384, F128, 35) (dual of [16384, 16315, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(1286, 20, F128, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,128)), using
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- Reed–Solomon code RS(122,128) [i]
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- construction X applied to Ce(34) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(12875, 16404, F128, 35) (dual of [16404, 16329, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(12875, 8202, F128, 2, 35) (dual of [(8202, 2), 16329, 36]-NRT-code), using
(40, 40+35, large)-Net in Base 128 — Upper bound on s
There is no (40, 75, large)-net in base 128, because
- 33 times m-reduction [i] would yield (40, 42, large)-net in base 128, but