Best Known (139−26, 139, s)-Nets in Base 4
(139−26, 139, 1262)-Net over F4 — Constructive and digital
Digital (113, 139, 1262)-net over F4, using
- net defined by OOA [i] based on linear OOA(4139, 1262, F4, 26, 26) (dual of [(1262, 26), 32673, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4139, 16406, F4, 26) (dual of [16406, 16267, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4139, 16410, F4, 26) (dual of [16410, 16271, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(4134, 16384, F4, 26) (dual of [16384, 16250, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4139, 16410, F4, 26) (dual of [16410, 16271, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4139, 16406, F4, 26) (dual of [16406, 16267, 27]-code), using
(139−26, 139, 9445)-Net over F4 — Digital
Digital (113, 139, 9445)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4139, 9445, F4, 26) (dual of [9445, 9306, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4139, 16410, F4, 26) (dual of [16410, 16271, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(4134, 16384, F4, 26) (dual of [16384, 16250, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4139, 16410, F4, 26) (dual of [16410, 16271, 27]-code), using
(139−26, 139, 5172431)-Net in Base 4 — Upper bound on s
There is no (113, 139, 5172432)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 485668 059256 102085 687915 517369 914941 224440 340846 018932 650809 376948 638828 120650 370655 > 4139 [i]