Best Known (84−12, 84, s)-Nets in Base 4
(84−12, 84, 43694)-Net over F4 — Constructive and digital
Digital (72, 84, 43694)-net over F4, using
- 1 times m-reduction [i] based on digital (72, 85, 43694)-net over F4, using
- net defined by OOA [i] based on linear OOA(485, 43694, F4, 13, 13) (dual of [(43694, 13), 567937, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(485, 262165, F4, 13) (dual of [262165, 262080, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, 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(485, 262165, F4, 13) (dual of [262165, 262080, 14]-code), using
- net defined by OOA [i] based on linear OOA(485, 43694, F4, 13, 13) (dual of [(43694, 13), 567937, 14]-NRT-code), using
(84−12, 84, 149945)-Net over F4 — Digital
Digital (72, 84, 149945)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(484, 149945, F4, 12) (dual of [149945, 149861, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(484, 262165, F4, 12) (dual of [262165, 262081, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(483, 262164, F4, 12) (dual of [262164, 262081, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(483, 262164, F4, 12) (dual of [262164, 262081, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(484, 262165, F4, 12) (dual of [262165, 262081, 13]-code), using
(84−12, 84, large)-Net in Base 4 — Upper bound on s
There is no (72, 84, large)-net in base 4, because
- 10 times m-reduction [i] would yield (72, 74, large)-net in base 4, but