Best Known (31, 36, s)-Nets in Base 4
(31, 36, 2097158)-Net over F4 — Constructive and digital
Digital (31, 36, 2097158)-net over F4, using
- 41 times duplication [i] based on digital (30, 35, 2097158)-net over F4, using
- net defined by OOA [i] based on linear OOA(435, 2097158, F4, 5, 5) (dual of [(2097158, 5), 10485755, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(435, 4194317, F4, 5) (dual of [4194317, 4194282, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(434, 4194304, F4, 5) (dual of [4194304, 4194270, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(423, 4194304, F4, 3) (dual of [4194304, 4194281, 4]-code or 4194304-cap in PG(22,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(412, 13, F4, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,4)), using
- dual of repetition code with length 13 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(435, 4194317, F4, 5) (dual of [4194317, 4194282, 6]-code), using
- net defined by OOA [i] based on linear OOA(435, 2097158, F4, 5, 5) (dual of [(2097158, 5), 10485755, 6]-NRT-code), using
(31, 36, 4194319)-Net over F4 — Digital
Digital (31, 36, 4194319)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(436, 4194319, F4, 5) (dual of [4194319, 4194283, 6]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(435, 4194317, F4, 5) (dual of [4194317, 4194282, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(434, 4194304, F4, 5) (dual of [4194304, 4194270, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(423, 4194304, F4, 3) (dual of [4194304, 4194281, 4]-code or 4194304-cap in PG(22,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(412, 13, F4, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,4)), using
- dual of repetition code with length 13 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(435, 4194318, F4, 4) (dual of [4194318, 4194283, 5]-code), using Gilbert–Varšamov bound and bm = 435 > Vbs−1(k−1) = 332 044322 434367 627536 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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(435, 4194317, F4, 5) (dual of [4194317, 4194282, 6]-code), using
- construction X with Varšamov bound [i] based on
(31, 36, large)-Net in Base 4 — Upper bound on s
There is no (31, 36, large)-net in base 4, because
- 3 times m-reduction [i] would yield (31, 33, large)-net in base 4, but