Best Known (58−8, 58, s)-Nets in Base 4
(58−8, 58, 65541)-Net over F4 — Constructive and digital
Digital (50, 58, 65541)-net over F4, using
- t-expansion [i] based on digital (49, 58, 65541)-net over F4, using
- net defined by OOA [i] based on linear OOA(458, 65541, F4, 9, 9) (dual of [(65541, 9), 589811, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(458, 262165, F4, 9) (dual of [262165, 262107, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(455, 262144, F4, 9) (dual of [262144, 262089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(437, 262144, F4, 6) (dual of [262144, 262107, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(458, 262165, F4, 9) (dual of [262165, 262107, 10]-code), using
- net defined by OOA [i] based on linear OOA(458, 65541, F4, 9, 9) (dual of [(65541, 9), 589811, 10]-NRT-code), using
(58−8, 58, 262168)-Net over F4 — Digital
Digital (50, 58, 262168)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(458, 262168, F4, 8) (dual of [262168, 262110, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(456, 262164, F4, 8) (dual of [262164, 262108, 9]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(455, 262144, F4, 9) (dual of [262144, 262089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(437, 262144, F4, 6) (dual of [262144, 262107, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(419, 20, F4, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,4)), using
- dual of repetition code with length 20 [i]
- linear OA(41, 20, F4, 1) (dual of [20, 19, 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 X4 applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(456, 262166, F4, 7) (dual of [262166, 262110, 8]-code), using Gilbert–Varšamov bound and bm = 456 > Vbs−1(k−1) = 328 716696 825394 241046 852791 503480 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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(456, 262164, F4, 8) (dual of [262164, 262108, 9]-code), using
- construction X with Varšamov bound [i] based on
(58−8, 58, large)-Net in Base 4 — Upper bound on s
There is no (50, 58, large)-net in base 4, because
- 6 times m-reduction [i] would yield (50, 52, large)-net in base 4, but