Best Known (64, 64+10, s)-Nets in Base 5
(64, 64+10, 390629)-Net over F5 — Constructive and digital
Digital (64, 74, 390629)-net over F5, using
- net defined by OOA [i] based on linear OOA(574, 390629, F5, 10, 10) (dual of [(390629, 10), 3906216, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(574, 1953145, F5, 10) (dual of [1953145, 1953071, 11]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(573, 1953125, F5, 11) (dual of [1953125, 1953052, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(519, 20, F5, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,5)), using
- dual of repetition code with length 20 [i]
- linear OA(51, 20, F5, 1) (dual of [20, 19, 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(10) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(574, 1953145, F5, 10) (dual of [1953145, 1953071, 11]-code), using
(64, 64+10, 1953145)-Net over F5 — Digital
Digital (64, 74, 1953145)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(574, 1953145, F5, 10) (dual of [1953145, 1953071, 11]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(573, 1953125, F5, 11) (dual of [1953125, 1953052, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(519, 20, F5, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,5)), using
- dual of repetition code with length 20 [i]
- linear OA(51, 20, F5, 1) (dual of [20, 19, 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(10) ⊂ Ce(7) [i] based on
(64, 64+10, large)-Net in Base 5 — Upper bound on s
There is no (64, 74, large)-net in base 5, because
- 8 times m-reduction [i] would yield (64, 66, large)-net in base 5, but