Best Known (53, 53+15, s)-Nets in Base 128
(53, 53+15, 1198650)-Net over F128 — Constructive and digital
Digital (53, 68, 1198650)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 279)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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 (1, 8, 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 (0, 3, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (42, 57, 1198371)-net over F128, using
- net defined by OOA [i] based on linear OOA(12857, 1198371, F128, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12857, 8388598, F128, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12857, large, F128, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12857, large, F128, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12857, 8388598, F128, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(12857, 1198371, F128, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- digital (4, 11, 279)-net over F128, using
(53, 53+15, 1220345)-Net in Base 128 — Constructive
(53, 68, 1220345)-net in base 128, using
- (u, u+v)-construction [i] based on
- (11, 18, 21974)-net in base 128, using
- net defined by OOA [i] based on OOA(12818, 21974, S128, 9, 7), using
- OOA stacking with additional row [i] based on OOA(12818, 21975, S128, 3, 7), using
- (u, u+v)-construction [i] based on
- linear OOA(1283, 129, F128, 3, 3) (dual of [(129, 3), 384, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;384,128) [i]
- OOA(12815, 21846, S128, 3, 7), using
- OOA 3-folding [i] based on OA(12815, 65538, S128, 7), using
- discarding parts of the base [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- OOA 3-folding [i] based on OA(12815, 65538, S128, 7), using
- linear OOA(1283, 129, F128, 3, 3) (dual of [(129, 3), 384, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on OOA(12818, 21975, S128, 3, 7), using
- net defined by OOA [i] based on OOA(12818, 21974, S128, 9, 7), using
- (35, 50, 1198371)-net in base 128, using
- net defined by OOA [i] based on OOA(12850, 1198371, S128, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12850, 8388598, S128, 15), using
- discarding factors based on OA(12850, large, S128, 15), using
- discarding parts of the base [i] based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding parts of the base [i] based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- discarding factors based on OA(12850, large, S128, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12850, 8388598, S128, 15), using
- net defined by OOA [i] based on OOA(12850, 1198371, S128, 15, 15), using
- (11, 18, 21974)-net in base 128, using
(53, 53+15, large)-Net over F128 — Digital
Digital (53, 68, large)-net over F128, using
- t-expansion [i] based on digital (51, 68, large)-net over F128, using
- 2 times m-reduction [i] based on digital (51, 70, large)-net over F128, using
(53, 53+15, large)-Net in Base 128 — Upper bound on s
There is no (53, 68, large)-net in base 128, because
- 13 times m-reduction [i] would yield (53, 55, large)-net in base 128, but