Best Known (78, 78+15, s)-Nets in Base 4
(78, 78+15, 9365)-Net over F4 — Constructive and digital
Digital (78, 93, 9365)-net over F4, using
- 41 times duplication [i] based on digital (77, 92, 9365)-net over F4, using
- net defined by OOA [i] based on linear OOA(492, 9365, F4, 15, 15) (dual of [(9365, 15), 140383, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(492, 65556, F4, 15) (dual of [65556, 65464, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(490, 65554, F4, 15) (dual of [65554, 65464, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 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(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(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(14) ⊂ Ce(12) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(490, 65554, F4, 15) (dual of [65554, 65464, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(492, 65556, F4, 15) (dual of [65556, 65464, 16]-code), using
- net defined by OOA [i] based on linear OOA(492, 9365, F4, 15, 15) (dual of [(9365, 15), 140383, 16]-NRT-code), using
(78, 78+15, 34427)-Net over F4 — Digital
Digital (78, 93, 34427)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(493, 34427, F4, 15) (dual of [34427, 34334, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(493, 65558, F4, 15) (dual of [65558, 65465, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- 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(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(465, 65536, F4, 11) (dual of [65536, 65471, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(41, 19, F4, 1) (dual of [19, 18, 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(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(493, 65558, F4, 15) (dual of [65558, 65465, 16]-code), using
(78, 78+15, large)-Net in Base 4 — Upper bound on s
There is no (78, 93, large)-net in base 4, because
- 13 times m-reduction [i] would yield (78, 80, large)-net in base 4, but