Best Known (90, 90+18, s)-Nets in Base 4
(90, 90+18, 7283)-Net over F4 — Constructive and digital
Digital (90, 108, 7283)-net over F4, using
- 41 times duplication [i] based on digital (89, 107, 7283)-net over F4, using
- net defined by OOA [i] based on linear OOA(4107, 7283, F4, 18, 18) (dual of [(7283, 18), 130987, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4107, 65547, F4, 18) (dual of [65547, 65440, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 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(17) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- OA 9-folding and stacking [i] based on linear OA(4107, 65547, F4, 18) (dual of [65547, 65440, 19]-code), using
- net defined by OOA [i] based on linear OOA(4107, 7283, F4, 18, 18) (dual of [(7283, 18), 130987, 19]-NRT-code), using
(90, 90+18, 32777)-Net over F4 — Digital
Digital (90, 108, 32777)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4108, 32777, F4, 2, 18) (dual of [(32777, 2), 65446, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4108, 65554, F4, 18) (dual of [65554, 65446, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4108, 65555, F4, 18) (dual of [65555, 65447, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(43, 19, F4, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4108, 65555, F4, 18) (dual of [65555, 65447, 19]-code), using
- OOA 2-folding [i] based on linear OA(4108, 65554, F4, 18) (dual of [65554, 65446, 19]-code), using
(90, 90+18, large)-Net in Base 4 — Upper bound on s
There is no (90, 108, large)-net in base 4, because
- 16 times m-reduction [i] would yield (90, 92, large)-net in base 4, but