Best Known (48, 48+7, s)-Nets in Base 4
(48, 48+7, 349545)-Net over F4 — Constructive and digital
Digital (48, 55, 349545)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 17)-net over F4, using
- net defined by OOA [i] based on linear OOA(44, 17, F4, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
- 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
- net defined by OOA [i] based on linear OOA(451, 349528, F4, 7, 7) (dual of [(349528, 7), 2446645, 8]-NRT-code), using
- digital (1, 4, 17)-net over F4, using
(48, 48+7, 1048604)-Net over F4 — Digital
Digital (48, 55, 1048604)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(455, 1048604, F4, 7) (dual of [1048604, 1048549, 8]-code), using
- (u, u+v)-construction [i] based on
- linear OA(43, 6, F4, 3) (dual of [6, 3, 4]-code or 6-arc in PG(2,4) or 6-cap in PG(2,4)), using
- linear OA(452, 1048598, F4, 7) (dual of [1048598, 1048546, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [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(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(421, 22, F4, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,4)), using
- dual of repetition code with length 22 [i]
- linear OA(41, 22, F4, 1) (dual of [22, 21, 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(6) ⊂ Ce(4) [i] based on
- (u, u+v)-construction [i] based on
(48, 48+7, large)-Net in Base 4 — Upper bound on s
There is no (48, 55, large)-net in base 4, because
- 5 times m-reduction [i] would yield (48, 50, large)-net in base 4, but