Best Known (248−33, 248, s)-Nets in Base 4
(248−33, 248, 65538)-Net over F4 — Constructive and digital
Digital (215, 248, 65538)-net over F4, using
- 42 times duplication [i] based on digital (213, 246, 65538)-net over F4, using
- net defined by OOA [i] based on linear OOA(4246, 65538, F4, 33, 33) (dual of [(65538, 33), 2162508, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4246, 1048609, F4, 33) (dual of [1048609, 1048363, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4246, 1048611, F4, 33) (dual of [1048611, 1048365, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- linear OA(4241, 1048576, F4, 33) (dual of [1048576, 1048335, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4246, 1048611, F4, 33) (dual of [1048611, 1048365, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4246, 1048609, F4, 33) (dual of [1048609, 1048363, 34]-code), using
- net defined by OOA [i] based on linear OOA(4246, 65538, F4, 33, 33) (dual of [(65538, 33), 2162508, 34]-NRT-code), using
(248−33, 248, 349537)-Net over F4 — Digital
Digital (215, 248, 349537)-net over F4, using
- 42 times duplication [i] based on digital (213, 246, 349537)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4246, 349537, F4, 3, 33) (dual of [(349537, 3), 1048365, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4246, 1048611, F4, 33) (dual of [1048611, 1048365, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- linear OA(4241, 1048576, F4, 33) (dual of [1048576, 1048335, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- OOA 3-folding [i] based on linear OA(4246, 1048611, F4, 33) (dual of [1048611, 1048365, 34]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4246, 349537, F4, 3, 33) (dual of [(349537, 3), 1048365, 34]-NRT-code), using
(248−33, 248, large)-Net in Base 4 — Upper bound on s
There is no (215, 248, large)-net in base 4, because
- 31 times m-reduction [i] would yield (215, 217, large)-net in base 4, but