Best Known (121, 121+27, s)-Nets in Base 5
(121, 121+27, 6010)-Net over F5 — Constructive and digital
Digital (121, 148, 6010)-net over F5, using
- net defined by OOA [i] based on linear OOA(5148, 6010, F5, 27, 27) (dual of [(6010, 27), 162122, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(5148, 78131, F5, 27) (dual of [78131, 77983, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(5148, 78132, F5, 27) (dual of [78132, 77984, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(5148, 78125, F5, 27) (dual of [78125, 77977, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(5141, 78125, F5, 26) (dual of [78125, 77984, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(5148, 78132, F5, 27) (dual of [78132, 77984, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(5148, 78131, F5, 27) (dual of [78131, 77983, 28]-code), using
(121, 121+27, 39066)-Net over F5 — Digital
Digital (121, 148, 39066)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5148, 39066, F5, 2, 27) (dual of [(39066, 2), 77984, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5148, 78132, F5, 27) (dual of [78132, 77984, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(5148, 78125, F5, 27) (dual of [78125, 77977, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(5141, 78125, F5, 26) (dual of [78125, 77984, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(5148, 78132, F5, 27) (dual of [78132, 77984, 28]-code), using
(121, 121+27, large)-Net in Base 5 — Upper bound on s
There is no (121, 148, large)-net in base 5, because
- 25 times m-reduction [i] would yield (121, 123, large)-net in base 5, but