Best Known (51−9, 51, s)-Nets in Base 4
(51−9, 51, 16386)-Net over F4 — Constructive and digital
Digital (42, 51, 16386)-net over F4, using
- 41 times duplication [i] based on digital (41, 50, 16386)-net over F4, using
- net defined by OOA [i] based on linear OOA(450, 16386, F4, 9, 9) (dual of [(16386, 9), 147424, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(450, 65545, F4, 9) (dual of [65545, 65495, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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 X applied to Ce(8) ⊂ Ce(6) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(450, 65545, F4, 9) (dual of [65545, 65495, 10]-code), using
- net defined by OOA [i] based on linear OOA(450, 16386, F4, 9, 9) (dual of [(16386, 9), 147424, 10]-NRT-code), using
(51−9, 51, 32773)-Net over F4 — Digital
Digital (42, 51, 32773)-net over F4, using
- 41 times duplication [i] based on digital (41, 50, 32773)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(450, 32773, F4, 2, 9) (dual of [(32773, 2), 65496, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(450, 65546, F4, 9) (dual of [65546, 65496, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(8) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(450, 65546, F4, 9) (dual of [65546, 65496, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(450, 32773, F4, 2, 9) (dual of [(32773, 2), 65496, 10]-NRT-code), using
(51−9, 51, large)-Net in Base 4 — Upper bound on s
There is no (42, 51, large)-net in base 4, because
- 7 times m-reduction [i] would yield (42, 44, large)-net in base 4, but