Best Known (106, 106+15, s)-Nets in Base 4
(106, 106+15, 149812)-Net over F4 — Constructive and digital
Digital (106, 121, 149812)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 15)-net over F4, using
- digital (96, 111, 149797)-net over F4, using
- net defined by OOA [i] based on linear OOA(4111, 149797, F4, 15, 15) (dual of [(149797, 15), 2246844, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4111, 1048580, F4, 15) (dual of [1048580, 1048469, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4111, 1048586, F4, 15) (dual of [1048586, 1048475, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(4101, 1048576, F4, 14) (dual of [1048576, 1048475, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4111, 1048586, F4, 15) (dual of [1048586, 1048475, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4111, 1048580, F4, 15) (dual of [1048580, 1048469, 16]-code), using
- net defined by OOA [i] based on linear OOA(4111, 149797, F4, 15, 15) (dual of [(149797, 15), 2246844, 16]-NRT-code), using
(106, 106+15, 681955)-Net over F4 — Digital
Digital (106, 121, 681955)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4121, 681955, F4, 15) (dual of [681955, 681834, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4121, 1048596, F4, 15) (dual of [1048596, 1048475, 16]-code), using
- (u, u+v)-construction [i] based on
- linear OA(410, 20, F4, 7) (dual of [20, 10, 8]-code), using
- extended quadratic residue code Qe(20,4) [i]
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(410, 20, F4, 7) (dual of [20, 10, 8]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4121, 1048596, F4, 15) (dual of [1048596, 1048475, 16]-code), using
(106, 106+15, large)-Net in Base 4 — Upper bound on s
There is no (106, 121, large)-net in base 4, because
- 13 times m-reduction [i] would yield (106, 108, large)-net in base 4, but