Best Known (202, 202+41, s)-Nets in Base 4
(202, 202+41, 3277)-Net over F4 — Constructive and digital
Digital (202, 243, 3277)-net over F4, using
- 41 times duplication [i] based on digital (201, 242, 3277)-net over F4, using
- net defined by OOA [i] based on linear OOA(4242, 3277, F4, 41, 41) (dual of [(3277, 41), 134115, 42]-NRT-code), using
- OOA 20-folding and stacking with additional row [i] based on linear OA(4242, 65541, F4, 41) (dual of [65541, 65299, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4242, 65545, F4, 41) (dual of [65545, 65303, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(38) [i] based on
- linear OA(4241, 65536, F4, 41) (dual of [65536, 65295, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4233, 65536, F4, 39) (dual of [65536, 65303, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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
- construction X applied to Ce(40) ⊂ Ce(38) [i] based on
- discarding factors / shortening the dual code based on linear OA(4242, 65545, F4, 41) (dual of [65545, 65303, 42]-code), using
- OOA 20-folding and stacking with additional row [i] based on linear OA(4242, 65541, F4, 41) (dual of [65541, 65299, 42]-code), using
- net defined by OOA [i] based on linear OOA(4242, 3277, F4, 41, 41) (dual of [(3277, 41), 134115, 42]-NRT-code), using
(202, 202+41, 32773)-Net over F4 — Digital
Digital (202, 243, 32773)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4243, 32773, F4, 2, 41) (dual of [(32773, 2), 65303, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4243, 65546, F4, 41) (dual of [65546, 65303, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4243, 65547, F4, 41) (dual of [65547, 65304, 42]-code), using
- construction XX applied to Ce(40) ⊂ Ce(38) ⊂ Ce(37) [i] based on
- linear OA(4241, 65536, F4, 41) (dual of [65536, 65295, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4233, 65536, F4, 39) (dual of [65536, 65303, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(4225, 65536, F4, 38) (dual of [65536, 65311, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(41, 10, F4, 1) (dual of [10, 9, 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(40) ⊂ Ce(38) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(4243, 65547, F4, 41) (dual of [65547, 65304, 42]-code), using
- OOA 2-folding [i] based on linear OA(4243, 65546, F4, 41) (dual of [65546, 65303, 42]-code), using
(202, 202+41, large)-Net in Base 4 — Upper bound on s
There is no (202, 243, large)-net in base 4, because
- 39 times m-reduction [i] would yield (202, 204, large)-net in base 4, but