MinT - Best Known (139, 153, s)-Nets in Base 8

Best Known (139, 153, s)-Nets in Base 8

(139, 153, 3095859)-Net over F₈ — Constructive and digital

Digital (139, 153, 3095859)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (40, 47, 699117)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁴⁷, 699117, F₈, 9, 7) (dual of [(699117, 9), 6292006, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(8⁴⁷, 699118, F₈, 3, 7) (dual of [(699118, 3), 2097307, 8]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(8⁴, 65, F₈, 3, 3) (dual of [(65, 3), 191, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(8⁴, 65, F₈, 2, 3) (dual of [(65, 2), 126, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
        ovoid in PG(3,Â 8) [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]

(139, 153, 3794845)-Net in Base 8 — Constructive

(139, 153, 3794845)-net in base 8, using

8¹ times duplication [i] based on (138, 152, 3794845)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (39, 46, 1398103)-net in base 8, using
    - net defined by OOA [i] based on OOA(8⁴⁶, 1398103, S₈, 7, 7), using
      - OOA 3-folding and stacking with additional row [i] based on OA(8⁴⁶, 4194310, S₈, 7), using
        trace code [i] based on OA(64²³, 2097155, S₆₄, 7), using
        discarding parts of the base [i] based on linear OA(128¹⁹, 2097155, F₁₂₈, 7) (dual of [2097155, 2097136, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(128¹⁹, 2097152, F₁₂₈, 7) (dual of [2097152, 2097133, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(128¹⁶, 2097152, F₁₂₈, 6) (dual of [2097152, 2097136, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 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]

(139, 153, large)-Net over F₈ — Digital

Digital (139, 153, large)-net over F₈, using

t-expansion [i] based on digital (138, 153, large)-net over F₈, using
- 8 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]

(139, 153, large)-Net in Base 8 — Upper bound on s

There is no (139, 153, large)-net in base 8, because

12 times m-reduction [i] would yield (139, 141, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (139, 153, s)-Nets in Base 8

(139, 153, 3095859)-Net over F8 — Constructive and digital

(139, 153, 3794845)-Net in Base 8 — Constructive

(139, 153, large)-Net over F8 — Digital

(139, 153, large)-Net in Base 8 — Upper bound on s

(139, 153, 3095859)-Net over F₈ — Constructive and digital

(139, 153, large)-Net over F₈ — Digital