Best Known (64−25, 64, s)-Nets in Base 128
(64−25, 64, 1557)-Net over F128 — Constructive and digital
Digital (39, 64, 1557)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- digital (24, 49, 1365)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 1365, F128, 25, 25) (dual of [(1365, 25), 34076, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12849, 16381, F128, 25) (dual of [16381, 16332, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12849, 16384, F128, 25) (dual of [16384, 16335, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(12849, 16384, F128, 25) (dual of [16384, 16335, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12849, 16381, F128, 25) (dual of [16381, 16332, 26]-code), using
- net defined by OOA [i] based on linear OOA(12849, 1365, F128, 25, 25) (dual of [(1365, 25), 34076, 26]-NRT-code), using
- digital (3, 15, 192)-net over F128, using
(64−25, 64, 5463)-Net in Base 128 — Constructive
(39, 64, 5463)-net in base 128, using
- base change [i] based on digital (31, 56, 5463)-net over F256, using
- net defined by OOA [i] based on linear OOA(25656, 5463, F256, 25, 25) (dual of [(5463, 25), 136519, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25656, 65557, F256, 25) (dual of [65557, 65501, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25656, 65560, F256, 25) (dual of [65560, 65504, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2567, 23, F256, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,256)), using
- discarding factors / shortening the dual code based on linear OA(2567, 256, F256, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
- Reed–Solomon code RS(249,256) [i]
- discarding factors / shortening the dual code based on linear OA(2567, 256, F256, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25656, 65560, F256, 25) (dual of [65560, 65504, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25656, 65557, F256, 25) (dual of [65557, 65501, 26]-code), using
- net defined by OOA [i] based on linear OOA(25656, 5463, F256, 25, 25) (dual of [(5463, 25), 136519, 26]-NRT-code), using
(64−25, 64, 32133)-Net over F128 — Digital
Digital (39, 64, 32133)-net over F128, using
(64−25, 64, large)-Net in Base 128 — Upper bound on s
There is no (39, 64, large)-net in base 128, because
- 23 times m-reduction [i] would yield (39, 41, large)-net in base 128, but