Best Known (58−14, 58, s)-Nets in Base 5
(58−14, 58, 448)-Net over F5 — Constructive and digital
Digital (44, 58, 448)-net over F5, using
- 51 times duplication [i] based on digital (43, 57, 448)-net over F5, using
- net defined by OOA [i] based on linear OOA(557, 448, F5, 14, 14) (dual of [(448, 14), 6215, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(557, 3136, F5, 14) (dual of [3136, 3079, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 X applied to Ce(13) ⊂ Ce(11) [i] based on
- OA 7-folding and stacking [i] based on linear OA(557, 3136, F5, 14) (dual of [3136, 3079, 15]-code), using
- net defined by OOA [i] based on linear OOA(557, 448, F5, 14, 14) (dual of [(448, 14), 6215, 15]-NRT-code), using
(58−14, 58, 2755)-Net over F5 — Digital
Digital (44, 58, 2755)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(558, 2755, F5, 14) (dual of [2755, 2697, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(558, 3138, F5, 14) (dual of [3138, 3080, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(541, 3125, F5, 11) (dual of [3125, 3084, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(51, 12, F5, 1) (dual of [12, 11, 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(13) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(558, 3138, F5, 14) (dual of [3138, 3080, 15]-code), using
(58−14, 58, 522781)-Net in Base 5 — Upper bound on s
There is no (44, 58, 522782)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 34694 561742 917305 378188 850303 781681 356185 > 558 [i]