Best Known (81, 81+10, s)-Nets in Base 4
(81, 81+10, 1677729)-Net over F4 — Constructive and digital
Digital (81, 91, 1677729)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (75, 85, 1677720)-net over F4, using
- net defined by OOA [i] based on linear OOA(485, 1677720, F4, 10, 10) (dual of [(1677720, 10), 16777115, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(485, 8388600, F4, 10) (dual of [8388600, 8388515, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(485, large, F4, 10) (dual of [large, large−85, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(485, large, F4, 10) (dual of [large, large−85, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(485, 8388600, F4, 10) (dual of [8388600, 8388515, 11]-code), using
- net defined by OOA [i] based on linear OOA(485, 1677720, F4, 10, 10) (dual of [(1677720, 10), 16777115, 11]-NRT-code), using
- digital (1, 6, 9)-net over F4, using
(81, 81+10, 7442921)-Net over F4 — Digital
Digital (81, 91, 7442921)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(491, 7442921, F4, 10) (dual of [7442921, 7442830, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(491, large, F4, 10) (dual of [large, large−91, 11]-code), using
- 6 times code embedding in larger space [i] based on linear OA(485, large, F4, 10) (dual of [large, large−85, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 6 times code embedding in larger space [i] based on linear OA(485, large, F4, 10) (dual of [large, large−85, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(491, large, F4, 10) (dual of [large, large−91, 11]-code), using
(81, 81+10, large)-Net in Base 4 — Upper bound on s
There is no (81, 91, large)-net in base 4, because
- 8 times m-reduction [i] would yield (81, 83, large)-net in base 4, but