Best Known (58, 58+12, s)-Nets in Base 4
(58, 58+12, 2739)-Net over F4 — Constructive and digital
Digital (58, 70, 2739)-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 (51, 63, 2730)-net over F4, using
- net defined by OOA [i] based on linear OOA(463, 2730, F4, 12, 12) (dual of [(2730, 12), 32697, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(463, 16380, F4, 12) (dual of [16380, 16317, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(463, 16383, F4, 12) (dual of [16383, 16320, 13]-code), using
- 1 times truncation [i] based on linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 1 times truncation [i] based on linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(463, 16383, F4, 12) (dual of [16383, 16320, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(463, 16380, F4, 12) (dual of [16380, 16317, 13]-code), using
- net defined by OOA [i] based on linear OOA(463, 2730, F4, 12, 12) (dual of [(2730, 12), 32697, 13]-NRT-code), using
- digital (1, 7, 9)-net over F4, using
(58, 58+12, 16413)-Net over F4 — Digital
Digital (58, 70, 16413)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(470, 16413, F4, 12) (dual of [16413, 16343, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(468, 16409, F4, 12) (dual of [16409, 16341, 13]-code), using
- 1 times truncation [i] based on linear OA(469, 16410, F4, 13) (dual of [16410, 16341, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-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
- 1 times truncation [i] based on linear OA(469, 16410, F4, 13) (dual of [16410, 16341, 14]-code), using
- linear OA(468, 16411, F4, 11) (dual of [16411, 16343, 12]-code), using Gilbert–Varšamov bound and bm = 468 > Vbs−1(k−1) = 22984 360908 873290 560057 037615 529707 133496 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(468, 16409, F4, 12) (dual of [16409, 16341, 13]-code), using
- construction X with Varšamov bound [i] based on
(58, 58+12, large)-Net in Base 4 — Upper bound on s
There is no (58, 70, large)-net in base 4, because
- 10 times m-reduction [i] would yield (58, 60, large)-net in base 4, but