Best Known (31, 31+10, s)-Nets in Base 5
(31, 31+10, 626)-Net over F5 — Constructive and digital
Digital (31, 41, 626)-net over F5, using
- net defined by OOA [i] based on linear OOA(541, 626, F5, 10, 10) (dual of [(626, 10), 6219, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(541, 3130, F5, 10) (dual of [3130, 3089, 11]-code), using
- 1 times truncation [i] based on linear OA(542, 3131, F5, 11) (dual of [3131, 3089, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- 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(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(51, 6, F5, 1) (dual of [6, 5, 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
- 1 times truncation [i] based on linear OA(542, 3131, F5, 11) (dual of [3131, 3089, 12]-code), using
- OA 5-folding and stacking [i] based on linear OA(541, 3130, F5, 10) (dual of [3130, 3089, 11]-code), using
(31, 31+10, 2936)-Net over F5 — Digital
Digital (31, 41, 2936)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(541, 2936, F5, 10) (dual of [2936, 2895, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(541, 3130, F5, 10) (dual of [3130, 3089, 11]-code), using
- 1 times truncation [i] based on linear OA(542, 3131, F5, 11) (dual of [3131, 3089, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- 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(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(51, 6, F5, 1) (dual of [6, 5, 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
- 1 times truncation [i] based on linear OA(542, 3131, F5, 11) (dual of [3131, 3089, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(541, 3130, F5, 10) (dual of [3130, 3089, 11]-code), using
(31, 31+10, 351015)-Net in Base 5 — Upper bound on s
There is no (31, 41, 351016)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 45475 070424 987531 890705 966497 > 541 [i]