Best Known (44, 44+7, s)-Nets in Base 4
(44, 44+7, 349528)-Net over F4 — Constructive and digital
Digital (44, 51, 349528)-net over F4, using
- net defined by OOA [i] based on linear OOA(451, 349528, F4, 7, 7) (dual of [(349528, 7), 2446645, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(451, 1048585, F4, 7) (dual of [1048585, 1048534, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(451, 1048586, F4, 7) (dual of [1048586, 1048535, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(441, 1048576, F4, 6) (dual of [1048576, 1048535, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(451, 1048586, F4, 7) (dual of [1048586, 1048535, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(451, 1048585, F4, 7) (dual of [1048585, 1048534, 8]-code), using
(44, 44+7, 910570)-Net over F4 — Digital
Digital (44, 51, 910570)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(451, 910570, F4, 7) (dual of [910570, 910519, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using
(44, 44+7, large)-Net in Base 4 — Upper bound on s
There is no (44, 51, large)-net in base 4, because
- 5 times m-reduction [i] would yield (44, 46, large)-net in base 4, but