Best Known (50, 50+21, s)-Nets in Base 128
(50, 50+21, 209844)-Net over F128 — Constructive and digital
Digital (50, 71, 209844)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (40, 61, 209715)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 209715, F128, 21, 21) (dual of [(209715, 21), 4403954, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12861, 2097151, F128, 21) (dual of [2097151, 2097090, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12861, 2097151, F128, 21) (dual of [2097151, 2097090, 22]-code), using
- net defined by OOA [i] based on linear OOA(12861, 209715, F128, 21, 21) (dual of [(209715, 21), 4403954, 22]-NRT-code), using
- digital (0, 10, 129)-net over F128, using
(50, 50+21, 838860)-Net in Base 128 — Constructive
(50, 71, 838860)-net in base 128, using
- 1281 times duplication [i] based on (49, 70, 838860)-net in base 128, using
- net defined by OOA [i] based on OOA(12870, 838860, S128, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12870, 8388601, S128, 21), using
- discarding factors based on OA(12870, large, S128, 21), using
- discarding parts of the base [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding parts of the base [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- discarding factors based on OA(12870, large, S128, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12870, 8388601, S128, 21), using
- net defined by OOA [i] based on OOA(12870, 838860, S128, 21, 21), using
(50, 50+21, 2097284)-Net over F128 — Digital
Digital (50, 71, 2097284)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12871, 2097284, F128, 21) (dual of [2097284, 2097213, 22]-code), using
- (u, u+v)-construction [i] based on
- linear OA(12810, 129, F128, 10) (dual of [129, 119, 11]-code or 129-arc in PG(9,128)), using
- extended Reed–Solomon code RSe(119,128) [i]
- linear OA(12861, 2097155, F128, 21) (dual of [2097155, 2097094, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(12810, 129, F128, 10) (dual of [129, 119, 11]-code or 129-arc in PG(9,128)), using
- (u, u+v)-construction [i] based on
(50, 50+21, large)-Net in Base 128 — Upper bound on s
There is no (50, 71, large)-net in base 128, because
- 19 times m-reduction [i] would yield (50, 52, large)-net in base 128, but