Best Known (124, 124+21, s)-Nets in Base 4
(124, 124+21, 26217)-Net over F4 — Constructive and digital
Digital (124, 145, 26217)-net over F4, using
- 44 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
(124, 124+21, 131094)-Net over F4 — Digital
Digital (124, 145, 131094)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4145, 131094, F4, 2, 21) (dual of [(131094, 2), 262043, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4145, 262188, F4, 21) (dual of [262188, 262043, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, 262189, F4, 21) (dual of [262189, 262044, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [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(4100, 262144, F4, 15) (dual of [262144, 262044, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(49, 45, F4, 5) (dual of [45, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4145, 262189, F4, 21) (dual of [262189, 262044, 22]-code), using
- OOA 2-folding [i] based on linear OA(4145, 262188, F4, 21) (dual of [262188, 262043, 22]-code), using
(124, 124+21, large)-Net in Base 4 — Upper bound on s
There is no (124, 145, large)-net in base 4, because
- 19 times m-reduction [i] would yield (124, 126, large)-net in base 4, but