Best Known (64−18, 64, s)-Nets in Base 128
(64−18, 64, 233209)-Net over F128 — Constructive and digital
Digital (46, 64, 233209)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 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 (34, 52, 233017)-net over F128, using
- net defined by OOA [i] based on linear OOA(12852, 233017, F128, 18, 18) (dual of [(233017, 18), 4194254, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12852, 2097153, F128, 18) (dual of [2097153, 2097101, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(12852, 2097153, F128, 18) (dual of [2097153, 2097101, 19]-code), using
- net defined by OOA [i] based on linear OOA(12852, 233017, F128, 18, 18) (dual of [(233017, 18), 4194254, 19]-NRT-code), using
- digital (3, 12, 192)-net over F128, using
(64−18, 64, 932067)-Net in Base 128 — Constructive
(46, 64, 932067)-net in base 128, using
- base change [i] based on digital (38, 56, 932067)-net over F256, using
- 2564 times duplication [i] based on digital (34, 52, 932067)-net over F256, using
- net defined by OOA [i] based on linear OOA(25652, 932067, F256, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- net defined by OOA [i] based on linear OOA(25652, 932067, F256, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
- 2564 times duplication [i] based on digital (34, 52, 932067)-net over F256, using
(64−18, 64, 4843597)-Net over F128 — Digital
Digital (46, 64, 4843597)-net over F128, using
(64−18, 64, 5062008)-Net in Base 128
(46, 64, 5062008)-net in base 128, using
- base change [i] based on digital (38, 56, 5062008)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25656, 5062008, F256, 18) (dual of [5062008, 5061952, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(25656, large, F256, 18) (dual of [large, large−56, 19]-code), using
- 4 times code embedding in larger space [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 4 times code embedding in larger space [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(25656, large, F256, 18) (dual of [large, large−56, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25656, 5062008, F256, 18) (dual of [5062008, 5061952, 19]-code), using
(64−18, 64, large)-Net in Base 128 — Upper bound on s
There is no (46, 64, large)-net in base 128, because
- 16 times m-reduction [i] would yield (46, 48, large)-net in base 128, but