Best Known (36, 36+20, s)-Nets in Base 9
(36, 36+20, 344)-Net over F9 — Constructive and digital
Digital (36, 56, 344)-net over F9, using
- 2 times m-reduction [i] based on digital (36, 58, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 29, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 29, 172)-net over F81, using
(36, 36+20, 751)-Net over F9 — Digital
Digital (36, 56, 751)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(956, 751, F9, 20) (dual of [751, 695, 21]-code), using
- 13 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0) [i] based on linear OA(953, 735, F9, 20) (dual of [735, 682, 21]-code), using
- construction XX applied to C1 = C([73,91]), C2 = C([75,92]), C3 = C1 + C2 = C([75,91]), and C∩ = C1 ∩ C2 = C([73,92]) [i] based on
- linear OA(949, 728, F9, 19) (dual of [728, 679, 20]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {73,74,…,91}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(949, 728, F9, 18) (dual of [728, 679, 19]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {75,76,…,92}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(952, 728, F9, 20) (dual of [728, 676, 21]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {73,74,…,92}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(946, 728, F9, 17) (dual of [728, 682, 18]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {75,76,…,91}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(91, 4, F9, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- Reed–Solomon code RS(8,9) [i]
- discarding factors / shortening the dual code based on linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- linear OA(90, 3, F9, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([73,91]), C2 = C([75,92]), C3 = C1 + C2 = C([75,91]), and C∩ = C1 ∩ C2 = C([73,92]) [i] based on
- 13 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0) [i] based on linear OA(953, 735, F9, 20) (dual of [735, 682, 21]-code), using
(36, 36+20, 124917)-Net in Base 9 — Upper bound on s
There is no (36, 56, 124918)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 273896 800769 957045 772869 759631 480500 861957 854002 362145 > 956 [i]