Best Known (20−5, 20, s)-Nets in Base 4
(20−5, 20, 2051)-Net over F4 — Constructive and digital
Digital (15, 20, 2051)-net over F4, using
- net defined by OOA [i] based on linear OOA(420, 2051, F4, 5, 5) (dual of [(2051, 5), 10235, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(420, 4103, F4, 5) (dual of [4103, 4083, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(413, 4096, F4, 3) (dual of [4096, 4083, 4]-code or 4096-cap in PG(12,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(2) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(420, 4103, F4, 5) (dual of [4103, 4083, 6]-code), using
(20−5, 20, 3937)-Net over F4 — Digital
Digital (15, 20, 3937)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(420, 3937, F4, 5) (dual of [3937, 3917, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(420, 4103, F4, 5) (dual of [4103, 4083, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(413, 4096, F4, 3) (dual of [4096, 4083, 4]-code or 4096-cap in PG(12,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(420, 4103, F4, 5) (dual of [4103, 4083, 6]-code), using
(20−5, 20, 247150)-Net in Base 4 — Upper bound on s
There is no (15, 20, 247151)-net in base 4, because
- 1 times m-reduction [i] would yield (15, 19, 247151)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 274878 870691 > 419 [i]