Best Known (46, 46+11, s)-Nets in Base 4
(46, 46+11, 3278)-Net over F4 — Constructive and digital
Digital (46, 57, 3278)-net over F4, using
- net defined by OOA [i] based on linear OOA(457, 3278, F4, 11, 11) (dual of [(3278, 11), 36001, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(457, 16391, F4, 11) (dual of [16391, 16334, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(457, 16384, F4, 11) (dual of [16384, 16327, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(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(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(457, 16391, F4, 11) (dual of [16391, 16334, 12]-code), using
(46, 46+11, 8195)-Net over F4 — Digital
Digital (46, 57, 8195)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(457, 8195, F4, 2, 11) (dual of [(8195, 2), 16333, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(457, 16390, F4, 11) (dual of [16390, 16333, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(457, 16391, F4, 11) (dual of [16391, 16334, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(457, 16384, F4, 11) (dual of [16384, 16327, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(457, 16391, F4, 11) (dual of [16391, 16334, 12]-code), using
- OOA 2-folding [i] based on linear OA(457, 16390, F4, 11) (dual of [16390, 16333, 12]-code), using
(46, 46+11, 4806030)-Net in Base 4 — Upper bound on s
There is no (46, 57, 4806031)-net in base 4, because
- 1 times m-reduction [i] would yield (46, 56, 4806031)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 5192 296963 080198 349083 610507 452682 > 456 [i]