Best Known (41, 41+7, s)-Nets in Base 5
(41, 41+7, 651045)-Net over F5 — Constructive and digital
Digital (41, 48, 651045)-net over F5, using
- 51 times duplication [i] based on digital (40, 47, 651045)-net over F5, using
- net defined by OOA [i] based on linear OOA(547, 651045, F5, 7, 7) (dual of [(651045, 7), 4557268, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(547, 1953136, F5, 7) (dual of [1953136, 1953089, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(510, 11, F5, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,5)), using
- dual of repetition code with length 11 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(547, 1953136, F5, 7) (dual of [1953136, 1953089, 8]-code), using
- net defined by OOA [i] based on linear OOA(547, 651045, F5, 7, 7) (dual of [(651045, 7), 4557268, 8]-NRT-code), using
(41, 41+7, 1953138)-Net over F5 — Digital
Digital (41, 48, 1953138)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(548, 1953138, F5, 7) (dual of [1953138, 1953090, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(547, 1953136, F5, 7) (dual of [1953136, 1953089, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(510, 11, F5, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,5)), using
- dual of repetition code with length 11 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 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(6) ⊂ Ce(5) [i] based on
- linear OA(547, 1953137, F5, 6) (dual of [1953137, 1953090, 7]-code), using Gilbert–Varšamov bound and bm = 547 > Vbs−1(k−1) = 242 537663 682937 878164 259189 631553 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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(547, 1953136, F5, 7) (dual of [1953136, 1953089, 8]-code), using
- construction X with Varšamov bound [i] based on
(41, 41+7, large)-Net in Base 5 — Upper bound on s
There is no (41, 48, large)-net in base 5, because
- 5 times m-reduction [i] would yield (41, 43, large)-net in base 5, but