Best Known (142, 142+41, s)-Nets in Base 4
(142, 142+41, 1044)-Net over F4 — Constructive and digital
Digital (142, 183, 1044)-net over F4, using
- 1 times m-reduction [i] based on digital (142, 184, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 46, 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, 46, 261)-net over F256, using
(142, 142+41, 3280)-Net over F4 — Digital
Digital (142, 183, 3280)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4183, 3280, F4, 41) (dual of [3280, 3097, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4183, 4105, F4, 41) (dual of [4105, 3922, 42]-code), using
- construction XX applied to Ce(40) ⊂ Ce(38) ⊂ Ce(37) [i] based on
- linear OA(4181, 4096, F4, 41) (dual of [4096, 3915, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4175, 4096, F4, 39) (dual of [4096, 3921, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- 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(41, 8, F4, 1) (dual of [8, 7, 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(40) ⊂ Ce(38) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(4183, 4105, F4, 41) (dual of [4105, 3922, 42]-code), using
(142, 142+41, 833532)-Net in Base 4 — Upper bound on s
There is no (142, 183, 833533)-net in base 4, because
- 1 times m-reduction [i] would yield (142, 182, 833533)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 37 577242 671268 151134 846350 994119 801429 157693 708867 322078 760790 798099 265938 804155 276917 396129 985721 910571 225499 > 4182 [i]