Best Known (117−14, 117, s)-Nets in Base 4
(117−14, 117, 599191)-Net over F4 — Constructive and digital
Digital (103, 117, 599191)-net over F4, using
- 41 times duplication [i] based on digital (102, 116, 599191)-net over F4, using
- net defined by OOA [i] based on linear OOA(4116, 599191, F4, 14, 14) (dual of [(599191, 14), 8388558, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4116, 4194337, F4, 14) (dual of [4194337, 4194221, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4116, 4194342, F4, 14) (dual of [4194342, 4194226, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(4111, 4194304, F4, 14) (dual of [4194304, 4194193, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(478, 4194304, F4, 10) (dual of [4194304, 4194226, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-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(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(4116, 4194342, F4, 14) (dual of [4194342, 4194226, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(4116, 4194337, F4, 14) (dual of [4194337, 4194221, 15]-code), using
- net defined by OOA [i] based on linear OOA(4116, 599191, F4, 14, 14) (dual of [(599191, 14), 8388558, 15]-NRT-code), using
(117−14, 117, 2097171)-Net over F4 — Digital
Digital (103, 117, 2097171)-net over F4, using
- 41 times duplication [i] based on digital (102, 116, 2097171)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4116, 2097171, F4, 2, 14) (dual of [(2097171, 2), 4194226, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4116, 4194342, F4, 14) (dual of [4194342, 4194226, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(4111, 4194304, F4, 14) (dual of [4194304, 4194193, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(478, 4194304, F4, 10) (dual of [4194304, 4194226, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-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(13) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(4116, 4194342, F4, 14) (dual of [4194342, 4194226, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4116, 2097171, F4, 2, 14) (dual of [(2097171, 2), 4194226, 15]-NRT-code), using
(117−14, 117, large)-Net in Base 4 — Upper bound on s
There is no (103, 117, large)-net in base 4, because
- 12 times m-reduction [i] would yield (103, 105, large)-net in base 4, but