Best Known (58−17, 58, s)-Nets in Base 128
(58−17, 58, 262294)-Net over F128 — Constructive and digital
Digital (41, 58, 262294)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 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 (32, 49, 262144)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 262144, F128, 17, 17) (dual of [(262144, 17), 4456399, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using
- net defined by OOA [i] based on linear OOA(12849, 262144, F128, 17, 17) (dual of [(262144, 17), 4456399, 18]-NRT-code), using
- digital (1, 9, 150)-net over F128, using
(58−17, 58, 1048575)-Net in Base 128 — Constructive
(41, 58, 1048575)-net in base 128, using
- 1282 times duplication [i] based on (39, 56, 1048575)-net in base 128, using
- base change [i] based on digital (32, 49, 1048575)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- base change [i] based on digital (32, 49, 1048575)-net over F256, using
(58−17, 58, 2330090)-Net over F128 — Digital
Digital (41, 58, 2330090)-net over F128, using
(58−17, 58, 2882188)-Net in Base 128
(41, 58, 2882188)-net in base 128, using
- 1282 times duplication [i] based on (39, 56, 2882188)-net in base 128, using
- base change [i] based on digital (32, 49, 2882188)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25649, 2882188, F256, 2, 17) (dual of [(2882188, 2), 5764327, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25649, 2882188, F256, 2, 17) (dual of [(2882188, 2), 5764327, 18]-NRT-code), using
- base change [i] based on digital (32, 49, 2882188)-net over F256, using
(58−17, 58, large)-Net in Base 128 — Upper bound on s
There is no (41, 58, large)-net in base 128, because
- 15 times m-reduction [i] would yield (41, 43, large)-net in base 128, but