MinT - Best Known (140, 154, s)-Nets in Base 8

Best Known (140, 154, s)-Nets in Base 8

(140, 154, 3096002)-Net over F₈ — Constructive and digital

Digital (140, 154, 3096002)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (41, 48, 699260)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁴⁸, 699260, F₈, 9, 7) (dual of [(699260, 9), 6293292, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(8⁴⁸, 699261, F₈, 3, 7) (dual of [(699261, 3), 2097735, 8]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(8⁵, 208, F₈, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(8⁵, 208, F₈, 2, 3) (dual of [(208, 2), 411, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁵, 208, F₈, 3) (dual of [208, 203, 4]-code or 208-cap in PG(4,8)), using
        caps completed using a computer search [i]
        linear OOA(8⁴³, 699053, F₈, 3, 7) (dual of [(699053, 3), 2097116, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(8⁴³, 2097159, F₈, 7) (dual of [2097159, 2097116, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(8⁴³, 2097152, F₈, 7) (dual of [2097152, 2097109, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(8³⁶, 2097152, F₈, 6) (dual of [2097152, 2097116, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(8⁰, 7, F₈, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (92, 106, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(140, 154, 3794849)-Net in Base 8 — Constructive

(140, 154, 3794849)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (41, 48, 1398107)-net in base 8, using
  - net defined by OOA [i] based on OOA(8⁴⁸, 1398107, S₈, 7, 7), using
    - OOA 3-folding and stacking with additional row [i] based on OA(8⁴⁸, 4194322, S₈, 7), using
      - trace code [i] based on OA(64²⁴, 2097161, S₆₄, 7), using
        discarding parts of the base [i] based on linear OA(128²⁰, 2097161, F₁₂₈, 7) (dual of [2097161, 2097141, 8]-code), using
        constructionÂ X4 applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
        linear OA(128¹⁹, 2097153, F₁₂₈, 7) (dual of [2097153, 2097134, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(128¹³, 2097153, F₁₂₈, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OA(128⁷, 8, F₁₂₈, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,128)), using
        dual of repetition code with length 8 [i]
        linear OA(128¹, 8, F₁₂₈, 1) (dual of [8, 7, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹, 128, F₁₂₈, 1) (dual of [128, 127, 2]-code), using
        Reedâ€“Solomon code RS(127,128) [i]
2. digital (92, 106, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(140, 154, large)-Net over F₈ — Digital

Digital (140, 154, large)-net over F₈, using

t-expansion [i] based on digital (138, 154, large)-net over F₈, using
- 7 times m-reduction [i] based on digital (138, 161, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶¹, large, F₈, 23) (dual of [large, large−161, 24]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(140, 154, large)-Net in Base 8 — Upper bound on s

There is no (140, 154, large)-net in base 8, because

12 times m-reduction [i] would yield (140, 142, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (140, 154, s)-Nets in Base 8

(140, 154, 3096002)-Net over F8 — Constructive and digital

(140, 154, 3794849)-Net in Base 8 — Constructive

(140, 154, large)-Net over F8 — Digital

(140, 154, large)-Net in Base 8 — Upper bound on s

(140, 154, 3096002)-Net over F₈ — Constructive and digital

(140, 154, large)-Net over F₈ — Digital