Best Known (41, 41+6, s)-Nets in Base 4
(41, 41+6, 1398106)-Net over F4 — Constructive and digital
Digital (41, 47, 1398106)-net over F4, using
- net defined by OOA [i] based on linear OOA(447, 1398106, F4, 6, 6) (dual of [(1398106, 6), 8388589, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(447, 4194318, F4, 6) (dual of [4194318, 4194271, 7]-code), using
- 1 times code embedding in larger space [i] based on linear OA(446, 4194317, F4, 6) (dual of [4194317, 4194271, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(445, 4194304, F4, 6) (dual of [4194304, 4194259, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- 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(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(5) ⊂ Ce(4) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(446, 4194317, F4, 6) (dual of [4194317, 4194271, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(447, 4194318, F4, 6) (dual of [4194318, 4194271, 7]-code), using
(41, 41+6, 4194319)-Net over F4 — Digital
Digital (41, 47, 4194319)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(447, 4194319, F4, 6) (dual of [4194319, 4194272, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(445, 4194315, F4, 6) (dual of [4194315, 4194270, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(445, 4194304, F4, 6) (dual of [4194304, 4194259, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- 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(40, 11, F4, 0) (dual of [11, 11, 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
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(445, 4194317, F4, 5) (dual of [4194317, 4194272, 6]-code), using Gilbert–Varšamov bound and bm = 445 > Vbs−1(k−1) = 1044 522699 534141 624369 067594 [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 (see above)
- linear OA(445, 4194315, F4, 6) (dual of [4194315, 4194270, 7]-code), using
- construction X with Varšamov bound [i] based on
(41, 41+6, large)-Net in Base 4 — Upper bound on s
There is no (41, 47, large)-net in base 4, because
- 4 times m-reduction [i] would yield (41, 43, large)-net in base 4, but