Best Known (58−11, 58, s)-Nets in Base 5
(58−11, 58, 15626)-Net over F5 — Constructive and digital
Digital (47, 58, 15626)-net over F5, using
- net defined by OOA [i] based on linear OOA(558, 15626, F5, 11, 11) (dual of [(15626, 11), 171828, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(558, 78131, F5, 11) (dual of [78131, 78073, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(558, 78133, F5, 11) (dual of [78133, 78075, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [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(550, 78125, F5, 9) (dual of [78125, 78075, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 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(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(558, 78133, F5, 11) (dual of [78133, 78075, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(558, 78131, F5, 11) (dual of [78131, 78073, 12]-code), using
(58−11, 58, 39066)-Net over F5 — Digital
Digital (47, 58, 39066)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(558, 39066, F5, 2, 11) (dual of [(39066, 2), 78074, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(558, 78132, F5, 11) (dual of [78132, 78074, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(558, 78133, F5, 11) (dual of [78133, 78075, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [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(550, 78125, F5, 9) (dual of [78125, 78075, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 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(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(558, 78133, F5, 11) (dual of [78133, 78075, 12]-code), using
- OOA 2-folding [i] based on linear OA(558, 78132, F5, 11) (dual of [78132, 78074, 12]-code), using
(58−11, 58, large)-Net in Base 5 — Upper bound on s
There is no (47, 58, large)-net in base 5, because
- 9 times m-reduction [i] would yield (47, 49, large)-net in base 5, but