Best Known (107, 132, s)-Nets in Base 4
(107, 132, 1367)-Net over F4 — Constructive and digital
Digital (107, 132, 1367)-net over F4, using
- net defined by OOA [i] based on linear OOA(4132, 1367, F4, 25, 25) (dual of [(1367, 25), 34043, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4132, 16405, F4, 25) (dual of [16405, 16273, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4132, 16410, F4, 25) (dual of [16410, 16278, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(24) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4132, 16410, F4, 25) (dual of [16410, 16278, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4132, 16405, F4, 25) (dual of [16405, 16273, 26]-code), using
(107, 132, 8424)-Net over F4 — Digital
Digital (107, 132, 8424)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4132, 8424, F4, 25) (dual of [8424, 8292, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4132, 16404, F4, 25) (dual of [16404, 16272, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(4127, 16385, F4, 25) (dual of [16385, 16258, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4113, 16385, F4, 21) (dual of [16385, 16272, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(45, 19, F4, 3) (dual of [19, 14, 4]-code or 19-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 C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4132, 16404, F4, 25) (dual of [16404, 16272, 26]-code), using
(107, 132, 6587608)-Net in Base 4 — Upper bound on s
There is no (107, 132, 6587609)-net in base 4, because
- 1 times m-reduction [i] would yield (107, 131, 6587609)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 7 410702 493289 001031 470256 140395 240939 958950 135230 313899 290048 410005 690613 852620 > 4131 [i]