Best Known (12, 12+9, s)-Nets in Base 4
(12, 12+9, 48)-Net over F4 — Constructive and digital
Digital (12, 21, 48)-net over F4, using
- 1 times m-reduction [i] based on digital (12, 22, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 11, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 11, 24)-net over F16, using
(12, 12+9, 55)-Net over F4 — Digital
Digital (12, 21, 55)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(421, 55, F4, 9) (dual of [55, 34, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(421, 70, F4, 9) (dual of [70, 49, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- linear OA(419, 64, F4, 9) (dual of [64, 45, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(416, 64, F4, 7) (dual of [64, 48, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(413, 64, F4, 6) (dual of [64, 51, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(41, 5, F4, 1) (dual of [5, 4, 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(8) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(421, 70, F4, 9) (dual of [70, 49, 10]-code), using
(12, 12+9, 752)-Net in Base 4 — Upper bound on s
There is no (12, 21, 753)-net in base 4, because
- 1 times m-reduction [i] would yield (12, 20, 753)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 101458 005156 > 420 [i]