MinT - Best Known (123, 123+13, s)-Nets in Base 9

Best Known (123, 123+13, s)-Nets in Base 9

(123, 123+13, 4390528)-Net over F₉ — Constructive and digital

Digital (123, 136, 4390528)-net over F₉, using

9¹ times duplication [i] based on digital (122, 135, 4390528)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (31, 37, 1594328)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9³⁷, 1594328, F₉, 6, 6) (dual of [(1594328, 6), 9565931, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(9³⁷, 4782984, F₉, 6) (dual of [4782984, 4782947, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(3) [i] based on
        linear OA(9³⁶, 4782969, F₉, 6) (dual of [4782969, 4782933, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(9²², 4782969, F₉, 4) (dual of [4782969, 4782947, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(9¹, 15, F₉, 1) (dual of [15, 14, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
  2. digital (85, 98, 2796200)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁹⁸, 2796200, F₉, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(9⁹⁸, 8388601, F₉, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(9⁹⁸, 8388602, F₉, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on linear OOA(81⁴⁹, 4194301, F₈₁, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁴⁹, 8388602, F₈₁, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴⁹, large, F₈₁, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(123, 123+13, large)-Net over F₉ — Digital

Digital (123, 136, large)-net over F₉, using

t-expansion [i] based on digital (117, 136, large)-net over F₉, using
- 1 times m-reduction [i] based on digital (117, 137, large)-net over F₉, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹³⁷, large, F₉, 20) (dual of [large, large−137, 21]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(123, 123+13, large)-Net in Base 9 — Upper bound on s

There is no (123, 136, large)-net in base 9, because

11 times m-reduction [i] would yield (123, 125, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (123, 123+13, s)-Nets in Base 9

(123, 123+13, 4390528)-Net over F9 — Constructive and digital

(123, 123+13, large)-Net over F9 — Digital

(123, 123+13, large)-Net in Base 9 — Upper bound on s

(123, 123+13, 4390528)-Net over F₉ — Constructive and digital

(123, 123+13, large)-Net over F₉ — Digital