Best Known (124−25, 124, s)-Nets in Base 4
(124−25, 124, 1062)-Net over F4 — Constructive and digital
Digital (99, 124, 1062)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (12, 24, 34)-net over F4, using
- trace code for nets [i] based on digital (0, 12, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- trace code for nets [i] based on digital (0, 12, 17)-net over F16, using
- digital (75, 100, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 25, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 25, 257)-net over F256, using
- digital (12, 24, 34)-net over F4, using
(124−25, 124, 4500)-Net over F4 — Digital
Digital (99, 124, 4500)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4124, 4500, F4, 25) (dual of [4500, 4376, 26]-code), using
- 388 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 11 times 0, 1, 18 times 0, 1, 29 times 0, 1, 42 times 0, 1, 61 times 0, 1, 86 times 0, 1, 117 times 0) [i] based on linear OA(4109, 4097, F4, 25) (dual of [4097, 3988, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 388 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 11 times 0, 1, 18 times 0, 1, 29 times 0, 1, 42 times 0, 1, 61 times 0, 1, 86 times 0, 1, 117 times 0) [i] based on linear OA(4109, 4097, F4, 25) (dual of [4097, 3988, 26]-code), using
(124−25, 124, 2614288)-Net in Base 4 — Upper bound on s
There is no (99, 124, 2614289)-net in base 4, because
- 1 times m-reduction [i] would yield (99, 123, 2614289)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 113 078536 002735 496349 763847 768756 662334 472637 393864 267628 994320 561645 296220 > 4123 [i]