Best Known (40, 40+7, s)-Nets in Base 4
(40, 40+7, 87387)-Net over F4 — Constructive and digital
Digital (40, 47, 87387)-net over F4, using
- net defined by OOA [i] based on linear OOA(447, 87387, F4, 7, 7) (dual of [(87387, 7), 611662, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(447, 262162, F4, 7) (dual of [262162, 262115, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(447, 262163, F4, 7) (dual of [262163, 262116, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(446, 262144, F4, 7) (dual of [262144, 262098, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(41, 19, F4, 1) (dual of [19, 18, 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 X applied to Ce(6) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(447, 262163, F4, 7) (dual of [262163, 262116, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(447, 262162, F4, 7) (dual of [262162, 262115, 8]-code), using
(40, 40+7, 262164)-Net over F4 — Digital
Digital (40, 47, 262164)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(447, 262164, F4, 7) (dual of [262164, 262117, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(446, 262144, F4, 7) (dual of [262144, 262098, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(419, 20, F4, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,4)), using
- dual of repetition code with length 20 [i]
- linear OA(41, 20, F4, 1) (dual of [20, 19, 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(6) ⊂ Ce(4) [i] based on
(40, 40+7, large)-Net in Base 4 — Upper bound on s
There is no (40, 47, large)-net in base 4, because
- 5 times m-reduction [i] would yield (40, 42, large)-net in base 4, but