Best Known (96−11, 96, s)-Nets in Base 4
(96−11, 96, 838878)-Net over F4 — Constructive and digital
Digital (85, 96, 838878)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 16)-net over F4, using
- digital (78, 89, 838862)-net over F4, using
- net defined by OOA [i] based on linear OOA(489, 838862, F4, 11, 11) (dual of [(838862, 11), 9227393, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(489, 4194311, F4, 11) (dual of [4194311, 4194222, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(489, 4194315, F4, 11) (dual of [4194315, 4194226, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(489, 4194304, F4, 11) (dual of [4194304, 4194215, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(40, 11, F4, 0) (dual of [11, 11, 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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(489, 4194315, F4, 11) (dual of [4194315, 4194226, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(489, 4194311, F4, 11) (dual of [4194311, 4194222, 12]-code), using
- net defined by OOA [i] based on linear OOA(489, 838862, F4, 11, 11) (dual of [(838862, 11), 9227393, 12]-NRT-code), using
(96−11, 96, 3131172)-Net over F4 — Digital
Digital (85, 96, 3131172)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(496, 3131172, F4, 11) (dual of [3131172, 3131076, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(496, 4194321, F4, 11) (dual of [4194321, 4194225, 12]-code), using
- (u, u+v)-construction [i] based on
- linear OA(47, 16, F4, 5) (dual of [16, 9, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 15 = 42−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(489, 4194305, F4, 11) (dual of [4194305, 4194216, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(47, 16, F4, 5) (dual of [16, 9, 6]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(496, 4194321, F4, 11) (dual of [4194321, 4194225, 12]-code), using
(96−11, 96, large)-Net in Base 4 — Upper bound on s
There is no (85, 96, large)-net in base 4, because
- 9 times m-reduction [i] would yield (85, 87, large)-net in base 4, but