Best Known (259−43, 259, s)-Nets in Base 4
(259−43, 259, 3121)-Net over F4 — Constructive and digital
Digital (216, 259, 3121)-net over F4, using
- 42 times duplication [i] based on digital (214, 257, 3121)-net over F4, using
- net defined by OOA [i] based on linear OOA(4257, 3121, F4, 43, 43) (dual of [(3121, 43), 133946, 44]-NRT-code), using
- OOA 21-folding and stacking with additional row [i] based on linear OA(4257, 65542, F4, 43) (dual of [65542, 65285, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4257, 65544, F4, 43) (dual of [65544, 65287, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(41) [i] based on
- linear OA(4257, 65536, F4, 43) (dual of [65536, 65279, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(4249, 65536, F4, 42) (dual of [65536, 65287, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(42) ⊂ Ce(41) [i] based on
- discarding factors / shortening the dual code based on linear OA(4257, 65544, F4, 43) (dual of [65544, 65287, 44]-code), using
- OOA 21-folding and stacking with additional row [i] based on linear OA(4257, 65542, F4, 43) (dual of [65542, 65285, 44]-code), using
- net defined by OOA [i] based on linear OOA(4257, 3121, F4, 43, 43) (dual of [(3121, 43), 133946, 44]-NRT-code), using
(259−43, 259, 33032)-Net over F4 — Digital
Digital (216, 259, 33032)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4259, 33032, F4, 43) (dual of [33032, 32773, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4259, 65555, F4, 43) (dual of [65555, 65296, 44]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4258, 65554, F4, 43) (dual of [65554, 65296, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,20]) [i] based on
- linear OA(4257, 65537, F4, 43) (dual of [65537, 65280, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(4241, 65537, F4, 41) (dual of [65537, 65296, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 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 C([0,21]) ⊂ C([0,20]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4258, 65554, F4, 43) (dual of [65554, 65296, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4259, 65555, F4, 43) (dual of [65555, 65296, 44]-code), using
(259−43, 259, large)-Net in Base 4 — Upper bound on s
There is no (216, 259, large)-net in base 4, because
- 41 times m-reduction [i] would yield (216, 218, large)-net in base 4, but