Best Known (53, 63, s)-Nets in Base 4
(53, 63, 13117)-Net over F4 — Constructive and digital
Digital (53, 63, 13117)-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 (47, 57, 13108)-net over F4, using
- net defined by OOA [i] based on linear OOA(457, 13108, F4, 10, 10) (dual of [(13108, 10), 131023, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(457, 65540, F4, 10) (dual of [65540, 65483, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(457, 65544, F4, 10) (dual of [65544, 65487, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(457, 65544, F4, 10) (dual of [65544, 65487, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(457, 65540, F4, 10) (dual of [65540, 65483, 11]-code), using
- net defined by OOA [i] based on linear OOA(457, 13108, F4, 10, 10) (dual of [(13108, 10), 131023, 11]-NRT-code), using
- digital (1, 6, 9)-net over F4, using
(53, 63, 58142)-Net over F4 — Digital
Digital (53, 63, 58142)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(463, 58142, F4, 10) (dual of [58142, 58079, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(463, 65548, F4, 10) (dual of [65548, 65485, 11]-code), using
- (u, u+v)-construction [i] based on
- linear OA(46, 12, F4, 5) (dual of [12, 6, 6]-code), using
- extended quadratic residue code Qe(12,4) [i]
- linear OA(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(46, 12, F4, 5) (dual of [12, 6, 6]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(463, 65548, F4, 10) (dual of [65548, 65485, 11]-code), using
(53, 63, large)-Net in Base 4 — Upper bound on s
There is no (53, 63, large)-net in base 4, because
- 8 times m-reduction [i] would yield (53, 55, large)-net in base 4, but