Best Known (75−12, 75, s)-Nets in Base 4
(75−12, 75, 10925)-Net over F4 — Constructive and digital
Digital (63, 75, 10925)-net over F4, using
- 1 times m-reduction [i] based on digital (63, 76, 10925)-net over F4, using
- net defined by OOA [i] based on linear OOA(476, 10925, F4, 13, 13) (dual of [(10925, 13), 141949, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(476, 65551, F4, 13) (dual of [65551, 65475, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(476, 65555, F4, 13) (dual of [65555, 65479, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(476, 65555, F4, 13) (dual of [65555, 65479, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(476, 65551, F4, 13) (dual of [65551, 65475, 14]-code), using
- net defined by OOA [i] based on linear OOA(476, 10925, F4, 13, 13) (dual of [(10925, 13), 141949, 14]-NRT-code), using
(75−12, 75, 43055)-Net over F4 — Digital
Digital (63, 75, 43055)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(475, 43055, F4, 12) (dual of [43055, 42980, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(475, 65555, F4, 12) (dual of [65555, 65480, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(474, 65554, F4, 12) (dual of [65554, 65480, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(417, 18, F4, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,4)), using
- dual of repetition code with length 18 [i]
- linear OA(41, 18, F4, 1) (dual of [18, 17, 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(12) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(474, 65554, F4, 12) (dual of [65554, 65480, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(475, 65555, F4, 12) (dual of [65555, 65480, 13]-code), using
(75−12, 75, large)-Net in Base 4 — Upper bound on s
There is no (63, 75, large)-net in base 4, because
- 10 times m-reduction [i] would yield (63, 65, large)-net in base 4, but