Best Known (122, 122+26, s)-Nets in Base 5
(122, 122+26, 6012)-Net over F5 — Constructive and digital
Digital (122, 148, 6012)-net over F5, using
- net defined by OOA [i] based on linear OOA(5148, 6012, F5, 26, 26) (dual of [(6012, 26), 156164, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(5148, 78156, F5, 26) (dual of [78156, 78008, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(5148, 78160, F5, 26) (dual of [78160, 78012, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(20) [i] based on
- 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(5113, 78125, F5, 21) (dual of [78125, 78012, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(57, 35, F5, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(25) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(5148, 78160, F5, 26) (dual of [78160, 78012, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(5148, 78156, F5, 26) (dual of [78156, 78008, 27]-code), using
(122, 122+26, 46810)-Net over F5 — Digital
Digital (122, 148, 46810)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5148, 46810, F5, 26) (dual of [46810, 46662, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(5148, 78155, F5, 26) (dual of [78155, 78007, 27]-code), using
- construction XX applied to Ce(25) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- 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(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(5113, 78125, F5, 21) (dual of [78125, 78012, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(55, 28, F5, 3) (dual of [28, 23, 4]-code or 28-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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 XX applied to Ce(25) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(5148, 78155, F5, 26) (dual of [78155, 78007, 27]-code), using
(122, 122+26, large)-Net in Base 5 — Upper bound on s
There is no (122, 148, large)-net in base 5, because
- 24 times m-reduction [i] would yield (122, 124, large)-net in base 5, but