Best Known (56−8, 56, s)-Nets in Base 5
(56−8, 56, 488286)-Net over F5 — Constructive and digital
Digital (48, 56, 488286)-net over F5, using
- net defined by OOA [i] based on linear OOA(556, 488286, F5, 8, 8) (dual of [(488286, 8), 3906232, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(556, 1953144, F5, 8) (dual of [1953144, 1953088, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(555, 1953125, F5, 8) (dual of [1953125, 1953070, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(51, 19, F5, 1) (dual of [19, 18, 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 X applied to Ce(7) ⊂ Ce(5) [i] based on
- OA 4-folding and stacking [i] based on linear OA(556, 1953144, F5, 8) (dual of [1953144, 1953088, 9]-code), using
(56−8, 56, 1911553)-Net over F5 — Digital
Digital (48, 56, 1911553)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(556, 1911553, F5, 8) (dual of [1911553, 1911497, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(556, 1953136, F5, 8) (dual of [1953136, 1953080, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(555, 1953125, F5, 8) (dual of [1953125, 1953070, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(556, 1953136, F5, 8) (dual of [1953136, 1953080, 9]-code), using
(56−8, 56, large)-Net in Base 5 — Upper bound on s
There is no (48, 56, large)-net in base 5, because
- 6 times m-reduction [i] would yield (48, 50, large)-net in base 5, but