Best Known (77, 77+22, s)-Nets in Base 4
(77, 77+22, 1036)-Net over F4 — Constructive and digital
Digital (77, 99, 1036)-net over F4, using
- 1 times m-reduction [i] based on digital (77, 100, 1036)-net over F4, using
- trace code for nets [i] based on digital (2, 25, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- trace code for nets [i] based on digital (2, 25, 259)-net over F256, using
(77, 77+22, 2452)-Net over F4 — Digital
Digital (77, 99, 2452)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(499, 2452, F4, 22) (dual of [2452, 2353, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(499, 4105, F4, 22) (dual of [4105, 4006, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- linear OA(497, 4096, F4, 22) (dual of [4096, 3999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(491, 4096, F4, 21) (dual of [4096, 4005, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(485, 4096, F4, 19) (dual of [4096, 4011, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(499, 4105, F4, 22) (dual of [4105, 4006, 23]-code), using
(77, 77+22, 428967)-Net in Base 4 — Upper bound on s
There is no (77, 99, 428968)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 401743 127152 534906 391516 847124 395672 746106 360351 225052 320580 > 499 [i]