Best Known (22, 22+12, s)-Nets in Base 9
(22, 22+12, 320)-Net over F9 — Constructive and digital
Digital (22, 34, 320)-net over F9, using
- trace code for nets [i] based on digital (5, 17, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
(22, 22+12, 753)-Net over F9 — Digital
Digital (22, 34, 753)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(934, 753, F9, 12) (dual of [753, 719, 13]-code), using
- 16 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0) [i] based on linear OA(931, 734, F9, 12) (dual of [734, 703, 13]-code), using
- construction XX applied to C1 = C([727,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([727,10]) [i] based on
- linear OA(928, 728, F9, 11) (dual of [728, 700, 12]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(928, 728, F9, 11) (dual of [728, 700, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(931, 728, F9, 12) (dual of [728, 697, 13]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(925, 728, F9, 10) (dual of [728, 703, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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
- linear OA(90, 3, F9, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([727,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([727,10]) [i] based on
- 16 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0) [i] based on linear OA(931, 734, F9, 12) (dual of [734, 703, 13]-code), using
(22, 22+12, 95607)-Net in Base 9 — Upper bound on s
There is no (22, 34, 95608)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 278 138687 598688 734188 743959 027585 > 934 [i]