Best Known (62−25, 62, s)-Nets in Base 128
(62−25, 62, 1515)-Net over F128 — Constructive and digital
Digital (37, 62, 1515)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-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 (1, 13, 150)-net over F128, using
(62−25, 62, 5462)-Net in Base 128 — Constructive
(37, 62, 5462)-net in base 128, using
- 1282 times duplication [i] based on (35, 60, 5462)-net in base 128, using
- net defined by OOA [i] based on OOA(12860, 5462, S128, 25, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(12860, 65545, S128, 25), using
- discarding factors based on OA(12860, 65548, S128, 25), using
- discarding parts of the base [i] based on linear OA(25652, 65548, F256, 25) (dual of [65548, 65496, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [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(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding parts of the base [i] based on linear OA(25652, 65548, F256, 25) (dual of [65548, 65496, 26]-code), using
- discarding factors based on OA(12860, 65548, S128, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(12860, 65545, S128, 25), using
- net defined by OOA [i] based on OOA(12860, 5462, S128, 25, 25), using
(62−25, 62, 21450)-Net over F128 — Digital
Digital (37, 62, 21450)-net over F128, using
(62−25, 62, large)-Net in Base 128 — Upper bound on s
There is no (37, 62, large)-net in base 128, because
- 23 times m-reduction [i] would yield (37, 39, large)-net in base 128, but