Best Known (48, 48+10, s)-Nets in Base 5
(48, 48+10, 15628)-Net over F5 — Constructive and digital
Digital (48, 58, 15628)-net over F5, using
- net defined by OOA [i] based on linear OOA(558, 15628, F5, 10, 10) (dual of [(15628, 10), 156222, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(558, 78140, F5, 10) (dual of [78140, 78082, 11]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 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(10) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(558, 78140, F5, 10) (dual of [78140, 78082, 11]-code), using
(48, 48+10, 78141)-Net over F5 — Digital
Digital (48, 58, 78141)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(558, 78141, F5, 10) (dual of [78141, 78083, 11]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(515, 16, F5, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,5)), using
- dual of repetition code with length 16 [i]
- linear OA(51, 16, F5, 1) (dual of [16, 15, 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
(48, 48+10, large)-Net in Base 5 — Upper bound on s
There is no (48, 58, large)-net in base 5, because
- 8 times m-reduction [i] would yield (48, 50, large)-net in base 5, but