Best Known (37, 37+33, s)-Nets in Base 128
(37, 37+33, 1025)-Net over F128 — Constructive and digital
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
(37, 37+33, 5652)-Net over F128 — Digital
Digital (37, 70, 5652)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12870, 5652, F128, 2, 33) (dual of [(5652, 2), 11234, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12870, 8201, F128, 2, 33) (dual of [(8201, 2), 16332, 34]-NRT-code), using
- OOA 2-folding [i] 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
- OOA 2-folding [i] based on linear OA(12870, 16402, F128, 33) (dual of [16402, 16332, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(12870, 8201, F128, 2, 33) (dual of [(8201, 2), 16332, 34]-NRT-code), using
(37, 37+33, large)-Net in Base 128 — Upper bound on s
There is no (37, 70, large)-net in base 128, because
- 31 times m-reduction [i] would yield (37, 39, large)-net in base 128, but