Best Known (81, 81+12, s)-Nets in Base 4
(81, 81+12, 174766)-Net over F4 — Constructive and digital
Digital (81, 93, 174766)-net over F4, using
- 1 times m-reduction [i] based on digital (81, 94, 174766)-net over F4, using
- net defined by OOA [i] based on linear OOA(494, 174766, F4, 13, 13) (dual of [(174766, 13), 2271864, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(494, 1048597, F4, 13) (dual of [1048597, 1048503, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(494, 1048597, F4, 13) (dual of [1048597, 1048503, 14]-code), using
- net defined by OOA [i] based on linear OOA(494, 174766, F4, 13, 13) (dual of [(174766, 13), 2271864, 14]-NRT-code), using
(81, 81+12, 524299)-Net over F4 — Digital
Digital (81, 93, 524299)-net over F4, using
- 41 times duplication [i] based on digital (80, 92, 524299)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(492, 524299, F4, 2, 12) (dual of [(524299, 2), 1048506, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(492, 1048598, F4, 12) (dual of [1048598, 1048506, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(421, 22, F4, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,4)), using
- dual of repetition code with length 22 [i]
- linear OA(41, 22, F4, 1) (dual of [22, 21, 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
- OOA 2-folding [i] based on linear OA(492, 1048598, F4, 12) (dual of [1048598, 1048506, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(492, 524299, F4, 2, 12) (dual of [(524299, 2), 1048506, 13]-NRT-code), using
(81, 81+12, large)-Net in Base 4 — Upper bound on s
There is no (81, 93, large)-net in base 4, because
- 10 times m-reduction [i] would yield (81, 83, large)-net in base 4, but