Best Known (52−15, 52, s)-Nets in Base 128
(52−15, 52, 299744)-Net over F128 — Constructive and digital
Digital (37, 52, 299744)-net over F128, using
- (u, u+v)-construction [i] based on
- 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 (29, 44, 299594)-net over F128, using
- net defined by OOA [i] based on linear OOA(12844, 299594, F128, 15, 15) (dual of [(299594, 15), 4493866, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12844, 2097159, F128, 15) (dual of [2097159, 2097115, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12844, 2097160, F128, 15) (dual of [2097160, 2097116, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12844, 2097160, F128, 15) (dual of [2097160, 2097116, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12844, 2097159, F128, 15) (dual of [2097159, 2097115, 16]-code), using
- net defined by OOA [i] based on linear OOA(12844, 299594, F128, 15, 15) (dual of [(299594, 15), 4493866, 16]-NRT-code), using
- digital (1, 8, 150)-net over F128, using
(52−15, 52, 1198371)-Net in Base 128 — Constructive
(37, 52, 1198371)-net in base 128, using
- 1282 times duplication [i] based on (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
(52−15, 52, 3194735)-Net over F128 — Digital
Digital (37, 52, 3194735)-net over F128, using
(52−15, 52, large)-Net in Base 128 — Upper bound on s
There is no (37, 52, large)-net in base 128, because
- 13 times m-reduction [i] would yield (37, 39, large)-net in base 128, but