Best Known (117−13, 117, s)-Nets in Base 4
(117−13, 117, 1398112)-Net over F4 — Constructive and digital
Digital (104, 117, 1398112)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 12)-net over F4, using
- digital (96, 109, 1398100)-net over F4, using
- net defined by OOA [i] based on linear OOA(4109, 1398100, F4, 13, 13) (dual of [(1398100, 13), 18175191, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4109, 8388601, F4, 13) (dual of [8388601, 8388492, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4109, 8388601, F4, 13) (dual of [8388601, 8388492, 14]-code), using
- net defined by OOA [i] based on linear OOA(4109, 1398100, F4, 13, 13) (dual of [(1398100, 13), 18175191, 14]-NRT-code), using
(117−13, 117, 4194313)-Net over F4 — Digital
Digital (104, 117, 4194313)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4117, 4194313, F4, 2, 13) (dual of [(4194313, 2), 8388509, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(48, 12, F4, 2, 6) (dual of [(12, 2), 16, 7]-NRT-code), using
- extracting embedded OOA [i] based on digital (2, 8, 12)-net over F4, using
- linear OOA(4109, 4194301, F4, 2, 13) (dual of [(4194301, 2), 8388493, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4109, 8388602, F4, 13) (dual of [8388602, 8388493, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- OOA 2-folding [i] based on linear OA(4109, 8388602, F4, 13) (dual of [8388602, 8388493, 14]-code), using
- linear OOA(48, 12, F4, 2, 6) (dual of [(12, 2), 16, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
(117−13, 117, large)-Net in Base 4 — Upper bound on s
There is no (104, 117, large)-net in base 4, because
- 11 times m-reduction [i] would yield (104, 106, large)-net in base 4, but