Best Known (209, 209+27, s)-Nets in Base 4
(209, 209+27, 322649)-Net over F4 — Constructive and digital
Digital (209, 236, 322649)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 15, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (194, 221, 322639)-net over F4, using
- net defined by OOA [i] based on linear OOA(4221, 322639, F4, 27, 27) (dual of [(322639, 27), 8711032, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4221, 4194308, F4, 27) (dual of [4194308, 4194087, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4221, 4194315, F4, 27) (dual of [4194315, 4194094, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4221, 4194304, F4, 27) (dual of [4194304, 4194083, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4210, 4194304, F4, 26) (dual of [4194304, 4194094, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 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(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4221, 4194315, F4, 27) (dual of [4194315, 4194094, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4221, 4194308, F4, 27) (dual of [4194308, 4194087, 28]-code), using
- net defined by OOA [i] based on linear OOA(4221, 322639, F4, 27, 27) (dual of [(322639, 27), 8711032, 28]-NRT-code), using
- digital (2, 15, 10)-net over F4, using
(209, 209+27, 2097193)-Net over F4 — Digital
Digital (209, 236, 2097193)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4236, 2097193, F4, 2, 27) (dual of [(2097193, 2), 4194150, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4236, 4194386, F4, 27) (dual of [4194386, 4194150, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- linear OA(4221, 4194305, F4, 27) (dual of [4194305, 4194084, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4155, 4194305, F4, 19) (dual of [4194305, 4194150, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(415, 81, F4, 7) (dual of [81, 66, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(415, 85, F4, 7) (dual of [85, 70, 8]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- OOA 2-folding [i] based on linear OA(4236, 4194386, F4, 27) (dual of [4194386, 4194150, 28]-code), using
(209, 209+27, large)-Net in Base 4 — Upper bound on s
There is no (209, 236, large)-net in base 4, because
- 25 times m-reduction [i] would yield (209, 211, large)-net in base 4, but