Best Known (74, 74+25, s)-Nets in Base 4
(74, 74+25, 531)-Net over F4 — Constructive and digital
Digital (74, 99, 531)-net over F4, using
- t-expansion [i] based on digital (73, 99, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 33, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 33, 177)-net over F64, using
(74, 74+25, 576)-Net in Base 4 — Constructive
(74, 99, 576)-net in base 4, using
- trace code for nets [i] based on (8, 33, 192)-net in base 64, using
- 2 times m-reduction [i] based on (8, 35, 192)-net in base 64, using
- base change [i] based on digital (3, 30, 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
- base change [i] based on digital (3, 30, 192)-net over F128, using
- 2 times m-reduction [i] based on (8, 35, 192)-net in base 64, using
(74, 74+25, 1083)-Net over F4 — Digital
Digital (74, 99, 1083)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(499, 1083, F4, 25) (dual of [1083, 984, 26]-code), using
- 50 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0) [i] based on linear OA(491, 1025, F4, 25) (dual of [1025, 934, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 50 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0) [i] based on linear OA(491, 1025, F4, 25) (dual of [1025, 934, 26]-code), using
(74, 74+25, 145557)-Net in Base 4 — Upper bound on s
There is no (74, 99, 145558)-net in base 4, because
- 1 times m-reduction [i] would yield (74, 98, 145558)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 100436 406314 667271 815341 897190 786157 197031 300624 723564 137370 > 498 [i]