Best Known (67, 67+25, s)-Nets in Base 4
(67, 67+25, 384)-Net over F4 — Constructive and digital
Digital (67, 92, 384)-net over F4, using
- 1 times m-reduction [i] based on digital (67, 93, 384)-net over F4, using
- trace code for nets [i] based on digital (5, 31, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 31, 128)-net over F64, using
(67, 67+25, 387)-Net in Base 4 — Constructive
(67, 92, 387)-net in base 4, using
- 1 times m-reduction [i] based on (67, 93, 387)-net in base 4, using
- trace code for nets [i] based on (5, 31, 129)-net in base 64, using
- 4 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
- base change [i] based on digital (0, 30, 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
- base change [i] based on digital (0, 30, 129)-net over F128, using
- 4 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
- trace code for nets [i] based on (5, 31, 129)-net in base 64, using
(67, 67+25, 741)-Net over F4 — Digital
Digital (67, 92, 741)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(492, 741, F4, 25) (dual of [741, 649, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(492, 1030, F4, 25) (dual of [1030, 938, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(491, 1024, F4, 25) (dual of [1024, 933, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(486, 1024, F4, 23) (dual of [1024, 938, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(41, 6, F4, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(492, 1030, F4, 25) (dual of [1030, 938, 26]-code), using
(67, 67+25, 64833)-Net in Base 4 — Upper bound on s
There is no (67, 92, 64834)-net in base 4, because
- 1 times m-reduction [i] would yield (67, 91, 64834)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 6 130775 721083 744947 446929 987527 981802 999816 943018 727120 > 491 [i]