MinT - Best Known (141−13, 141, s)-Nets in Base 9

Best Known (141−13, 141, s)-Nets in Base 9

(141−13, 141, 5592400)-Net over F₉ — Constructive and digital

Digital (128, 141, 5592400)-net over F₉, using

9¹ times duplication [i] based on digital (127, 140, 5592400)-net over F₉, using
- trace code for nets [i] based on digital (57, 70, 2796200)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁷⁰, 2796200, F₈₁, 14, 13) (dual of [(2796200, 14), 39146730, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(81⁷⁰, 8388601, F₈₁, 2, 13) (dual of [(8388601, 2), 16777132, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(81⁷⁰, 8388602, F₈₁, 2, 13) (dual of [(8388602, 2), 16777134, 14]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(81²¹, 4194301, F₈₁, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81²¹, 8388602, F₈₁, 6) (dual of [8388602, 8388581, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(81²¹, large, F₈₁, 6) (dual of [large, large−21, 7]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        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]

(141−13, 141, 5592402)-Net in Base 9 — Constructive

(128, 141, 5592402)-net in base 9, using

9¹ times duplication [i] based on (127, 140, 5592402)-net in base 9, using
- (u,Â u+v)-construction [i] based on
  1. (33, 39, 2796201)-net in base 9, using
    - base change [i] based on digital (20, 26, 2796201)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27²⁶, 2796201, F₂₇, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(27²⁶, large, F₂₇, 6) (dual of [large, large−26, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  2. digital (88, 101, 2796201)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9¹⁰¹, 2796201, F₉, 15, 13) (dual of [(2796201, 15), 41942914, 14]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OOA(9¹⁰¹, 5592403, F₉, 3, 13) (dual of [(5592403, 3), 16777108, 14]-NRT-code), using
        1 times NRT-code embedding in larger space [i] based on linear OOA(9⁹⁸, 5592402, F₉, 3, 13) (dual of [(5592402, 3), 16777108, 14]-NRT-code), using
        trace code [i] based on linear OOA(81⁴⁹, 2796201, F₈₁, 3, 13) (dual of [(2796201, 3), 8388554, 14]-NRT-code), using
        OOA 3-folding [i] 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]

(141−13, 141, large)-Net over F₉ — Digital

Digital (128, 141, large)-net over F₉, using

t-expansion [i] based on digital (124, 141, large)-net over F₉, using
- 4 times m-reduction [i] based on digital (124, 145, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(141−13, 141, large)-Net in Base 9 — Upper bound on s

There is no (128, 141, large)-net in base 9, because

11 times m-reduction [i] would yield (128, 130, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (141−13, 141, s)-Nets in Base 9

(141−13, 141, 5592400)-Net over F9 — Constructive and digital

(141−13, 141, 5592402)-Net in Base 9 — Constructive

(141−13, 141, large)-Net over F9 — Digital

(141−13, 141, large)-Net in Base 9 — Upper bound on s

(141−13, 141, 5592400)-Net over F₉ — Constructive and digital

(141−13, 141, large)-Net over F₉ — Digital