Best Known (239−37, 239, s)-Nets in Base 4
(239−37, 239, 3655)-Net over F4 — Constructive and digital
Digital (202, 239, 3655)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 22, 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 (180, 217, 3640)-net over F4, using
- net defined by OOA [i] based on linear OOA(4217, 3640, F4, 37, 37) (dual of [(3640, 37), 134463, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4217, 65521, F4, 37) (dual of [65521, 65304, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- discarding factors / shortening the dual code based on linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4217, 65521, F4, 37) (dual of [65521, 65304, 38]-code), using
- net defined by OOA [i] based on linear OOA(4217, 3640, F4, 37, 37) (dual of [(3640, 37), 134463, 38]-NRT-code), using
- digital (4, 22, 15)-net over F4, using
(239−37, 239, 57536)-Net over F4 — Digital
Digital (202, 239, 57536)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4239, 57536, F4, 37) (dual of [57536, 57297, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4239, 65560, F4, 37) (dual of [65560, 65321, 38]-code), using
- (u, u+v)-construction [i] based on
- linear OA(422, 24, F4, 18) (dual of [24, 2, 19]-code), using
- 1 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
- 1 times truncation [i] based on linear OA(423, 25, F4, 19) (dual of [25, 2, 20]-code), using
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(422, 24, F4, 18) (dual of [24, 2, 19]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4239, 65560, F4, 37) (dual of [65560, 65321, 38]-code), using
(239−37, 239, large)-Net in Base 4 — Upper bound on s
There is no (202, 239, large)-net in base 4, because
- 35 times m-reduction [i] would yield (202, 204, large)-net in base 4, but