Best Known (105, 105+30, s)-Nets in Base 4
(105, 105+30, 1040)-Net over F4 — Constructive and digital
Digital (105, 135, 1040)-net over F4, using
- 1 times m-reduction [i] based on digital (105, 136, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 34, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- trace code for nets [i] based on digital (3, 34, 260)-net over F256, using
(105, 105+30, 2843)-Net over F4 — Digital
Digital (105, 135, 2843)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4135, 2843, F4, 30) (dual of [2843, 2708, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4135, 4105, F4, 30) (dual of [4105, 3970, 31]-code), using
- construction XX applied to Ce(29) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- linear OA(4133, 4096, F4, 30) (dual of [4096, 3963, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4127, 4096, F4, 29) (dual of [4096, 3969, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4121, 4096, F4, 27) (dual of [4096, 3975, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(29) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4135, 4105, F4, 30) (dual of [4105, 3970, 31]-code), using
(105, 105+30, 561275)-Net in Base 4 — Upper bound on s
There is no (105, 135, 561276)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1897 167031 457696 869578 387058 312823 502083 193761 376758 352515 203935 218144 433471 757444 > 4135 [i]