Best Known (34−6, 34, s)-Nets in Base 4
(34−6, 34, 21848)-Net over F4 — Constructive and digital
Digital (28, 34, 21848)-net over F4, using
- 41 times duplication [i] based on digital (27, 33, 21848)-net over F4, using
- net defined by OOA [i] based on linear OOA(433, 21848, F4, 6, 6) (dual of [(21848, 6), 131055, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(433, 65544, F4, 6) (dual of [65544, 65511, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(433, 65544, F4, 6) (dual of [65544, 65511, 7]-code), using
- net defined by OOA [i] based on linear OOA(433, 21848, F4, 6, 6) (dual of [(21848, 6), 131055, 7]-NRT-code), using
(34−6, 34, 65546)-Net over F4 — Digital
Digital (28, 34, 65546)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(434, 65546, F4, 6) (dual of [65546, 65512, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(49, 10, F4, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,4)), using
- dual of repetition code with length 10 [i]
- linear OA(41, 10, F4, 1) (dual of [10, 9, 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(5) ⊂ Ce(4) [i] based on
(34−6, 34, 4032820)-Net in Base 4 — Upper bound on s
There is no (28, 34, 4032821)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 295 148103 991941 877444 > 434 [i]