Best Known (79−11, 79, s)-Nets in Base 4
(79−11, 79, 52439)-Net over F4 — Constructive and digital
Digital (68, 79, 52439)-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 (62, 73, 52430)-net over F4, using
- net defined by OOA [i] based on linear OOA(473, 52430, F4, 11, 11) (dual of [(52430, 11), 576657, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(473, 262151, F4, 11) (dual of [262151, 262078, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(473, 262153, F4, 11) (dual of [262153, 262080, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(473, 262144, F4, 11) (dual of [262144, 262071, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 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(473, 262153, F4, 11) (dual of [262153, 262080, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(473, 262151, F4, 11) (dual of [262151, 262078, 12]-code), using
- net defined by OOA [i] based on linear OOA(473, 52430, F4, 11, 11) (dual of [(52430, 11), 576657, 12]-NRT-code), using
- digital (1, 6, 9)-net over F4, using
(79−11, 79, 228282)-Net over F4 — Digital
Digital (68, 79, 228282)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(479, 228282, F4, 11) (dual of [228282, 228203, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(479, 262157, F4, 11) (dual of [262157, 262078, 12]-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(473, 262145, F4, 11) (dual of [262145, 262072, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [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(479, 262157, F4, 11) (dual of [262157, 262078, 12]-code), using
(79−11, 79, large)-Net in Base 4 — Upper bound on s
There is no (68, 79, large)-net in base 4, because
- 9 times m-reduction [i] would yield (68, 70, large)-net in base 4, but