Best Known (86, 86+25, s)-Nets in Base 4
(86, 86+25, 1036)-Net over F4 — Constructive and digital
Digital (86, 111, 1036)-net over F4, using
- 1 times m-reduction [i] based on digital (86, 112, 1036)-net over F4, using
- trace code for nets [i] based on digital (2, 28, 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, 28, 259)-net over F256, using
(86, 86+25, 2363)-Net over F4 — Digital
Digital (86, 111, 2363)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4111, 2363, F4, 25) (dual of [2363, 2252, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4111, 4105, F4, 25) (dual of [4105, 3994, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4109, 4096, F4, 25) (dual of [4096, 3987, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4103, 4096, F4, 23) (dual of [4096, 3993, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4111, 4105, F4, 25) (dual of [4105, 3994, 26]-code), using
(86, 86+25, 582259)-Net in Base 4 — Upper bound on s
There is no (86, 111, 582260)-net in base 4, because
- 1 times m-reduction [i] would yield (86, 110, 582260)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 685025 951132 918695 722354 158652 921479 485351 623932 508925 255312 170780 > 4110 [i]