Best Known (44, 44+8, s)-Nets in Base 5
(44, 44+8, 97661)-Net over F5 — Constructive and digital
Digital (44, 52, 97661)-net over F5, using
- 51 times duplication [i] based on digital (43, 51, 97661)-net over F5, using
- net defined by OOA [i] based on linear OOA(551, 97661, F5, 8, 8) (dual of [(97661, 8), 781237, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(551, 390644, F5, 8) (dual of [390644, 390593, 9]-code), using
- 1 times code embedding in larger space [i] based on linear OA(550, 390643, F5, 8) (dual of [390643, 390593, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(533, 390625, F5, 6) (dual of [390625, 390592, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(517, 18, F5, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,5)), using
- dual of repetition code with length 18 [i]
- linear OA(51, 18, F5, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(550, 390643, F5, 8) (dual of [390643, 390593, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(551, 390644, F5, 8) (dual of [390644, 390593, 9]-code), using
- net defined by OOA [i] based on linear OOA(551, 97661, F5, 8, 8) (dual of [(97661, 8), 781237, 9]-NRT-code), using
(44, 44+8, 390647)-Net over F5 — Digital
Digital (44, 52, 390647)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(552, 390647, F5, 8) (dual of [390647, 390595, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(550, 390643, F5, 8) (dual of [390643, 390593, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(533, 390625, F5, 6) (dual of [390625, 390592, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(517, 18, F5, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,5)), using
- dual of repetition code with length 18 [i]
- linear OA(51, 18, F5, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(550, 390645, F5, 7) (dual of [390645, 390595, 8]-code), using Gilbert–Varšamov bound and bm = 550 > Vbs−1(k−1) = 20216 193814 027964 445822 360679 409329 [i]
- linear OA(50, 2, F5, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(550, 390643, F5, 8) (dual of [390643, 390593, 9]-code), using
- construction X with Varšamov bound [i] based on
(44, 44+8, large)-Net in Base 5 — Upper bound on s
There is no (44, 52, large)-net in base 5, because
- 6 times m-reduction [i] would yield (44, 46, large)-net in base 5, but