Best Known (133, 171, s)-Nets in Base 4
(133, 171, 1044)-Net over F4 — Constructive and digital
Digital (133, 171, 1044)-net over F4, using
- 1 times m-reduction [i] based on digital (133, 172, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 43, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- trace code for nets [i] based on digital (4, 43, 261)-net over F256, using
(133, 171, 3288)-Net over F4 — Digital
Digital (133, 171, 3288)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4171, 3288, F4, 38) (dual of [3288, 3117, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4171, 4105, F4, 38) (dual of [4105, 3934, 39]-code), using
- construction XX applied to Ce(37) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4169, 4096, F4, 38) (dual of [4096, 3927, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-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(37) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4171, 4105, F4, 38) (dual of [4105, 3934, 39]-code), using
(133, 171, 692826)-Net in Base 4 — Upper bound on s
There is no (133, 171, 692827)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 8 959073 795717 183785 450348 367112 970312 611195 872711 592815 727114 209377 923301 925697 874127 939555 925382 412236 > 4171 [i]