Best Known (89−13, 89, s)-Nets in Base 4
(89−13, 89, 43699)-Net over F4 — Constructive and digital
Digital (76, 89, 43699)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (69, 82, 43690)-net over F4, using
- net defined by OOA [i] based on linear OOA(482, 43690, F4, 13, 13) (dual of [(43690, 13), 567888, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(482, 262141, F4, 13) (dual of [262141, 262059, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(482, 262141, F4, 13) (dual of [262141, 262059, 14]-code), using
- net defined by OOA [i] based on linear OOA(482, 43690, F4, 13, 13) (dual of [(43690, 13), 567888, 14]-NRT-code), using
- digital (1, 7, 9)-net over F4, using
(89−13, 89, 131089)-Net over F4 — Digital
Digital (76, 89, 131089)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(489, 131089, F4, 2, 13) (dual of [(131089, 2), 262089, 14]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(487, 131088, F4, 2, 13) (dual of [(131088, 2), 262089, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(487, 262176, F4, 13) (dual of [262176, 262089, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(455, 262144, F4, 9) (dual of [262144, 262089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-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(12) ⊂ Ce(8) [i] based on
- OOA 2-folding [i] based on linear OA(487, 262176, F4, 13) (dual of [262176, 262089, 14]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(487, 131088, F4, 2, 13) (dual of [(131088, 2), 262089, 14]-NRT-code), using
(89−13, 89, large)-Net in Base 4 — Upper bound on s
There is no (76, 89, large)-net in base 4, because
- 11 times m-reduction [i] would yield (76, 78, large)-net in base 4, but