Best Known (40, 40+33, s)-Nets in Base 128
(40, 40+33, 1025)-Net over F128 — Constructive and digital
Digital (40, 73, 1025)-net over F128, using
- 1283 times duplication [i] based on digital (37, 70, 1025)-net over F128, using
- net defined by OOA [i] based on linear OOA(12870, 1025, F128, 33, 33) (dual of [(1025, 33), 33755, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(12870, 16401, F128, 33) (dual of [16401, 16331, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(12870, 16402, F128, 33) (dual of [16402, 16332, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- 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(12853, 16385, F128, 27) (dual of [16385, 16332, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12870, 16402, F128, 33) (dual of [16402, 16332, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(12870, 16401, F128, 33) (dual of [16401, 16331, 34]-code), using
- net defined by OOA [i] based on linear OOA(12870, 1025, F128, 33, 33) (dual of [(1025, 33), 33755, 34]-NRT-code), using
(40, 40+33, 8205)-Net over F128 — Digital
Digital (40, 73, 8205)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12873, 8205, F128, 2, 33) (dual of [(8205, 2), 16337, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12873, 16410, F128, 33) (dual of [16410, 16337, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(23) [i] based on
- linear OA(12865, 16384, F128, 33) (dual of [16384, 16319, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(1288, 26, F128, 8) (dual of [26, 18, 9]-code or 26-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(32) ⊂ Ce(23) [i] based on
- OOA 2-folding [i] based on linear OA(12873, 16410, F128, 33) (dual of [16410, 16337, 34]-code), using
(40, 40+33, large)-Net in Base 128 — Upper bound on s
There is no (40, 73, large)-net in base 128, because
- 31 times m-reduction [i] would yield (40, 42, large)-net in base 128, but