Best Known (46, 46+10, s)-Nets in Base 4
(46, 46+10, 3287)-Net over F4 — Constructive and digital
Digital (46, 56, 3287)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (40, 50, 3278)-net over F4, using
- net defined by OOA [i] based on linear OOA(450, 3278, F4, 10, 10) (dual of [(3278, 10), 32730, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(450, 16390, F4, 10) (dual of [16390, 16340, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(450, 16391, F4, 10) (dual of [16391, 16341, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(450, 16384, F4, 10) (dual of [16384, 16334, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(450, 16391, F4, 10) (dual of [16391, 16341, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(450, 16390, F4, 10) (dual of [16390, 16340, 11]-code), using
- net defined by OOA [i] based on linear OOA(450, 3278, F4, 10, 10) (dual of [(3278, 10), 32730, 11]-NRT-code), using
- digital (1, 6, 9)-net over F4, using
(46, 46+10, 16412)-Net over F4 — Digital
Digital (46, 56, 16412)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(456, 16412, F4, 10) (dual of [16412, 16356, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(5) ⊂ Ce(4) [i] based on
- linear OA(450, 16384, F4, 10) (dual of [16384, 16334, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(9) ⊂ Ce(5) ⊂ Ce(4) [i] based on
(46, 46+10, 4806030)-Net in Base 4 — Upper bound on s
There is no (46, 56, 4806031)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 5192 296963 080198 349083 610507 452682 > 456 [i]