Best Known (248−30, 248, s)-Nets in Base 4
(248−30, 248, 279622)-Net over F4 — Constructive and digital
Digital (218, 248, 279622)-net over F4, using
- 41 times duplication [i] based on digital (217, 247, 279622)-net over F4, using
- net defined by OOA [i] based on linear OOA(4247, 279622, F4, 30, 30) (dual of [(279622, 30), 8388413, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4247, 4194330, F4, 30) (dual of [4194330, 4194083, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- linear OA(4243, 4194304, F4, 30) (dual of [4194304, 4194061, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4221, 4194304, F4, 27) (dual of [4194304, 4194083, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(44, 26, F4, 2) (dual of [26, 22, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- Hamming code H(4,4) [i]
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- OA 15-folding and stacking [i] based on linear OA(4247, 4194330, F4, 30) (dual of [4194330, 4194083, 31]-code), using
- net defined by OOA [i] based on linear OOA(4247, 279622, F4, 30, 30) (dual of [(279622, 30), 8388413, 31]-NRT-code), using
(248−30, 248, 1398114)-Net over F4 — Digital
Digital (218, 248, 1398114)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4248, 1398114, F4, 3, 30) (dual of [(1398114, 3), 4194094, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4248, 4194342, F4, 30) (dual of [4194342, 4194094, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(25) [i] based on
- linear OA(4243, 4194304, F4, 30) (dual of [4194304, 4194061, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4210, 4194304, F4, 26) (dual of [4194304, 4194094, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-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(29) ⊂ Ce(25) [i] based on
- OOA 3-folding [i] based on linear OA(4248, 4194342, F4, 30) (dual of [4194342, 4194094, 31]-code), using
(248−30, 248, large)-Net in Base 4 — Upper bound on s
There is no (218, 248, large)-net in base 4, because
- 28 times m-reduction [i] would yield (218, 220, large)-net in base 4, but