MinT - Best Known (114−11, 114, s)-Nets in Base 8

Best Known (114−11, 114, s)-Nets in Base 8

(114−11, 114, 4404092)-Net over F₈ — Constructive and digital

Digital (103, 114, 4404092)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (27, 32, 1048652)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 3, 73)-net over F₈, using
      - kÂ =Â 2 construction [i]
    2. digital (24, 29, 1048579)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8²⁹, 1048579, F₈, 5, 5) (dual of [(1048579, 5), 5242866, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(8²⁹, 2097159, F₈, 5) (dual of [2097159, 2097130, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(8²⁹, 2097152, F₈, 5) (dual of [2097152, 2097123, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(8²², 2097152, F₈, 4) (dual of [2097152, 2097130, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [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 (71, 82, 3355440)-net over F₈, using
  - trace code for nets [i] based on digital (30, 41, 1677720)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(114−11, 114, 5452594)-Net in Base 8 — Constructive

(103, 114, 5452594)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (27, 32, 2097154)-net in base 8, using
  - net defined by OOA [i] based on OOA(8³², 2097154, S₈, 5, 5), using
    - OOA 2-folding and stacking with additional row [i] based on OA(8³², 4194309, S₈, 5), using
      - discarding factors based on OA(8³², 4194310, S₈, 5), using
        trace code [i] based on OA(64¹⁶, 2097155, S₆₄, 5), using
        discarding parts of the base [i] based on linear OA(128¹³, 2097155, F₁₂₈, 5) (dual of [2097155, 2097142, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(128¹³, 2097152, F₁₂₈, 5) (dual of [2097152, 2097139, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(128¹⁰, 2097152, F₁₂₈, 4) (dual of [2097152, 2097142, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [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 (71, 82, 3355440)-net over F₈, using
  - trace code for nets [i] based on digital (30, 41, 1677720)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(114−11, 114, large)-Net over F₈ — Digital

Digital (103, 114, large)-net over F₈, using

t-expansion [i] based on digital (100, 114, large)-net over F₈, using
- 3 times m-reduction [i] based on digital (100, 117, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁷, large, F₈, 17) (dual of [large, large−117, 18]-code), using
    - 4 times code embedding in larger space [i] based on linear OA(8¹¹³, large, F₈, 17) (dual of [large, large−113, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(114−11, 114, large)-Net in Base 8 — Upper bound on s

There is no (103, 114, large)-net in base 8, because

9 times m-reduction [i] would yield (103, 105, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (114−11, 114, s)-Nets in Base 8

(114−11, 114, 4404092)-Net over F8 — Constructive and digital

(114−11, 114, 5452594)-Net in Base 8 — Constructive

(114−11, 114, large)-Net over F8 — Digital

(114−11, 114, large)-Net in Base 8 — Upper bound on s

(114−11, 114, 4404092)-Net over F₈ — Constructive and digital

(114−11, 114, large)-Net over F₈ — Digital