Best Known (237−53, 237, s)-Nets in Base 4
(237−53, 237, 1052)-Net over F4 — Constructive and digital
Digital (184, 237, 1052)-net over F4, using
- 41 times duplication [i] based on digital (183, 236, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 59, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
- trace code for nets [i] based on digital (6, 59, 263)-net over F256, using
(237−53, 237, 4003)-Net over F4 — Digital
Digital (184, 237, 4003)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4237, 4003, F4, 53) (dual of [4003, 3766, 54]-code), using
- discarding factors / shortening the dual code based on linear OA(4237, 4105, F4, 53) (dual of [4105, 3868, 54]-code), using
- construction XX applied to Ce(52) ⊂ Ce(50) ⊂ Ce(49) [i] based on
- linear OA(4235, 4096, F4, 53) (dual of [4096, 3861, 54]-code), using an extension Ce(52) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,52], and designed minimum distance d ≥ |I|+1 = 53 [i]
- linear OA(4229, 4096, F4, 51) (dual of [4096, 3867, 52]-code), using an extension Ce(50) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,50], and designed minimum distance d ≥ |I|+1 = 51 [i]
- linear OA(4223, 4096, F4, 50) (dual of [4096, 3873, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [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(52) ⊂ Ce(50) ⊂ Ce(49) [i] based on
- discarding factors / shortening the dual code based on linear OA(4237, 4105, F4, 53) (dual of [4105, 3868, 54]-code), using
(237−53, 237, 1025687)-Net in Base 4 — Upper bound on s
There is no (184, 237, 1025688)-net in base 4, because
- 1 times m-reduction [i] would yield (184, 236, 1025688)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 12194 493046 824648 485965 952724 644186 052264 059413 481975 211168 466188 540227 200470 469566 291093 320234 789173 740527 784995 435748 734806 563218 603725 313200 > 4236 [i]