Best Known (93−25, 93, s)-Nets in Base 4
(93−25, 93, 384)-Net over F4 — Constructive and digital
Digital (68, 93, 384)-net over F4, using
- t-expansion [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
(93−25, 93, 450)-Net in Base 4 — Constructive
(68, 93, 450)-net in base 4, using
- trace code for nets [i] based on (6, 31, 150)-net in base 64, using
- 4 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
- 4 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
(93−25, 93, 788)-Net over F4 — Digital
Digital (68, 93, 788)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(493, 788, F4, 25) (dual of [788, 695, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(493, 1032, F4, 25) (dual of [1032, 939, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(22) ⊂ Ce(21) [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(481, 1024, F4, 22) (dual of [1024, 943, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(493, 1032, F4, 25) (dual of [1032, 939, 26]-code), using
(93−25, 93, 72774)-Net in Base 4 — Upper bound on s
There is no (68, 93, 72775)-net in base 4, because
- 1 times m-reduction [i] would yield (68, 92, 72775)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 24 523637 782555 129091 635663 954130 387921 118654 873169 173016 > 492 [i]