Best Known (213, 213+34, s)-Nets in Base 4
(213, 213+34, 15435)-Net over F4 — Constructive and digital
Digital (213, 247, 15435)-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 (192, 226, 15420)-net over F4, using
- net defined by OOA [i] based on linear OOA(4226, 15420, F4, 34, 34) (dual of [(15420, 34), 524054, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4226, 262140, F4, 34) (dual of [262140, 261914, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4226, 262144, F4, 34) (dual of [262144, 261918, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- discarding factors / shortening the dual code based on linear OA(4226, 262144, F4, 34) (dual of [262144, 261918, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(4226, 262140, F4, 34) (dual of [262140, 261914, 35]-code), using
- net defined by OOA [i] based on linear OOA(4226, 15420, F4, 34, 34) (dual of [(15420, 34), 524054, 35]-NRT-code), using
- digital (4, 21, 15)-net over F4, using
(213, 213+34, 181148)-Net over F4 — Digital
Digital (213, 247, 181148)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4247, 181148, F4, 34) (dual of [181148, 180901, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4247, 262167, F4, 34) (dual of [262167, 261920, 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(4226, 262144, F4, 34) (dual of [262144, 261918, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, 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(4247, 262167, F4, 34) (dual of [262167, 261920, 35]-code), using
(213, 213+34, large)-Net in Base 4 — Upper bound on s
There is no (213, 247, large)-net in base 4, because
- 32 times m-reduction [i] would yield (213, 215, large)-net in base 4, but