Best Known (48, 59, s)-Nets in Base 5
(48, 59, 15626)-Net over F5 — Constructive and digital
Digital (48, 59, 15626)-net over F5, using
- 51 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(558, 15626, F5, 11, 11) (dual of [(15626, 11), 171828, 12]-NRT-code), using
(48, 59, 39067)-Net over F5 — Digital
Digital (48, 59, 39067)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(559, 39067, F5, 2, 11) (dual of [(39067, 2), 78075, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(559, 78134, F5, 11) (dual of [78134, 78075, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(559, 78135, F5, 11) (dual of [78135, 78076, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(8) ⊂ 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(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(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, 9, F5, 1) (dual of [9, 8, 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
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(10) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(559, 78135, F5, 11) (dual of [78135, 78076, 12]-code), using
- OOA 2-folding [i] based on linear OA(559, 78134, F5, 11) (dual of [78134, 78075, 12]-code), using
(48, 59, large)-Net in Base 5 — Upper bound on s
There is no (48, 59, large)-net in base 5, because
- 9 times m-reduction [i] would yield (48, 50, large)-net in base 5, but