Best Known (78−21, 78, s)-Nets in Base 4
(78−21, 78, 384)-Net over F4 — Constructive and digital
Digital (57, 78, 384)-net over F4, using
- trace code for nets [i] based on digital (5, 26, 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
(78−21, 78, 450)-Net in Base 4 — Constructive
(57, 78, 450)-net in base 4, using
- trace code for nets [i] based on (5, 26, 150)-net in base 64, using
- 2 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
- base change [i] based on digital (1, 24, 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
- base change [i] based on digital (1, 24, 150)-net over F128, using
- 2 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
(78−21, 78, 714)-Net over F4 — Digital
Digital (57, 78, 714)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(478, 714, F4, 21) (dual of [714, 636, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(478, 1032, F4, 21) (dual of [1032, 954, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- linear OA(476, 1024, F4, 21) (dual of [1024, 948, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(471, 1024, F4, 19) (dual of [1024, 953, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(466, 1024, F4, 18) (dual of [1024, 958, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(478, 1032, F4, 21) (dual of [1032, 954, 22]-code), using
(78−21, 78, 65262)-Net in Base 4 — Upper bound on s
There is no (57, 78, 65263)-net in base 4, because
- 1 times m-reduction [i] would yield (57, 77, 65263)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 22837 105846 632962 633440 749575 282326 248291 781573 > 477 [i]