Best Known (222−34, 222, s)-Nets in Base 4
(222−34, 222, 3870)-Net over F4 — Constructive and digital
Digital (188, 222, 3870)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 21, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- digital (167, 201, 3855)-net over F4, using
- net defined by OOA [i] based on linear OOA(4201, 3855, F4, 34, 34) (dual of [(3855, 34), 130869, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4201, 65535, F4, 34) (dual of [65535, 65334, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- discarding factors / shortening the dual code based on linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(4201, 65535, F4, 34) (dual of [65535, 65334, 35]-code), using
- net defined by OOA [i] based on linear OOA(4201, 3855, F4, 34, 34) (dual of [(3855, 34), 130869, 35]-NRT-code), using
- digital (4, 21, 15)-net over F4, using
(222−34, 222, 61313)-Net over F4 — Digital
Digital (188, 222, 61313)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4222, 61313, F4, 34) (dual of [61313, 61091, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4222, 65559, F4, 34) (dual of [65559, 65337, 35]-code), using
- (u, u+v)-construction [i] based on
- linear OA(421, 23, F4, 17) (dual of [23, 2, 18]-code), using
- 2 times truncation [i] based on linear OA(423, 25, F4, 19) (dual of [25, 2, 20]-code), using
- repeating each code word 5 times [i] based on linear OA(43, 5, F4, 3) (dual of [5, 2, 4]-code or 5-arc in PG(2,4) or 5-cap in PG(2,4)), using
- extended Reed–Solomon code RSe(2,4) [i]
- Simplex code S(2,4) [i]
- repeating each code word 5 times [i] based on linear OA(43, 5, F4, 3) (dual of [5, 2, 4]-code or 5-arc in PG(2,4) or 5-cap in PG(2,4)), using
- 2 times truncation [i] based on linear OA(423, 25, F4, 19) (dual of [25, 2, 20]-code), using
- linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(421, 23, F4, 17) (dual of [23, 2, 18]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4222, 65559, F4, 34) (dual of [65559, 65337, 35]-code), using
(222−34, 222, large)-Net in Base 4 — Upper bound on s
There is no (188, 222, large)-net in base 4, because
- 32 times m-reduction [i] would yield (188, 190, large)-net in base 4, but