Best Known (53, 53+11, s)-Nets in Base 4
(53, 53+11, 3294)-Net over F4 — Constructive and digital
Digital (53, 64, 3294)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 16)-net over F4, using
- 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
- net defined by OOA [i] based on linear OOA(457, 3278, F4, 11, 11) (dual of [(3278, 11), 36001, 12]-NRT-code), using
(53, 53+11, 16419)-Net over F4 — Digital
Digital (53, 64, 16419)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(464, 16419, F4, 11) (dual of [16419, 16355, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [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(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(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
(53, 53+11, large)-Net in Base 4 — Upper bound on s
There is no (53, 64, large)-net in base 4, because
- 9 times m-reduction [i] would yield (53, 55, large)-net in base 4, but