Best Known (122, 122+21, s)-Nets in Base 4
(122, 122+21, 26217)-Net over F4 — Constructive and digital
Digital (122, 143, 26217)-net over F4, using
- 42 times duplication [i] based on digital (120, 141, 26217)-net over F4, using
- net defined by OOA [i] based on linear OOA(4141, 26217, F4, 21, 21) (dual of [(26217, 21), 550416, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4141, 262171, F4, 21) (dual of [262171, 262030, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4141, 262176, F4, 21) (dual of [262176, 262035, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4141, 262176, F4, 21) (dual of [262176, 262035, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4141, 262171, F4, 21) (dual of [262171, 262030, 22]-code), using
- net defined by OOA [i] based on linear OOA(4141, 26217, F4, 21, 21) (dual of [(26217, 21), 550416, 22]-NRT-code), using
(122, 122+21, 130948)-Net over F4 — Digital
Digital (122, 143, 130948)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4143, 130948, F4, 2, 21) (dual of [(130948, 2), 261753, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4143, 131089, F4, 2, 21) (dual of [(131089, 2), 262035, 22]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4141, 131088, F4, 2, 21) (dual of [(131088, 2), 262035, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4141, 262176, F4, 21) (dual of [262176, 262035, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(4141, 262176, F4, 21) (dual of [262176, 262035, 22]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4141, 131088, F4, 2, 21) (dual of [(131088, 2), 262035, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4143, 131089, F4, 2, 21) (dual of [(131089, 2), 262035, 22]-NRT-code), using
(122, 122+21, large)-Net in Base 4 — Upper bound on s
There is no (122, 143, large)-net in base 4, because
- 19 times m-reduction [i] would yield (122, 124, large)-net in base 4, but