Best Known (66, 66+23, s)-Nets in Base 4
(66, 66+23, 384)-Net over F4 — Constructive and digital
Digital (66, 89, 384)-net over F4, using
- t-expansion [i] based on digital (65, 89, 384)-net over F4, using
- 1 times m-reduction [i] based on digital (65, 90, 384)-net over F4, using
- trace code for nets [i] based on digital (5, 30, 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, 30, 128)-net over F64, using
- 1 times m-reduction [i] based on digital (65, 90, 384)-net over F4, using
(66, 66+23, 450)-Net in Base 4 — Constructive
(66, 89, 450)-net in base 4, using
- 1 times m-reduction [i] based on (66, 90, 450)-net in base 4, using
- trace code for nets [i] based on (6, 30, 150)-net in base 64, using
- 5 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
- base change [i] based on digital (1, 30, 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, 30, 150)-net over F128, using
- 5 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
- trace code for nets [i] based on (6, 30, 150)-net in base 64, using
(66, 66+23, 949)-Net over F4 — Digital
Digital (66, 89, 949)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(489, 949, F4, 23) (dual of [949, 860, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(489, 1038, F4, 23) (dual of [1038, 949, 24]-code), using
- construction XX applied to Ce(22) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- 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(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(41, 12, F4, 1) (dual of [12, 11, 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(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(22) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(489, 1038, F4, 23) (dual of [1038, 949, 24]-code), using
(66, 66+23, 107235)-Net in Base 4 — Upper bound on s
There is no (66, 89, 107236)-net in base 4, because
- 1 times m-reduction [i] would yield (66, 88, 107236)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 95787 935342 473857 932147 215476 772560 996059 739320 369932 > 488 [i]