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Some combinatorial aspects of perfect codes. rs tt rst s r s t r t st

  1. Some combinatorial aspects of perfect codes. ❈❧❛✉❞✐♦ ◗✉r❡s❤✐ ❙t❛t❡ ❯♥✐✈❡rs✐t② ♦❢ ❈❛♠♣✐♥❛s✱ ❇r❛③✐❧ ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❙✳❈♦st❛ ❙♣❡❝✐❛❧ ❉❛②s ♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❈♦♥str✉❝t✐♦♥s ✉s✐♥❣ ❋✐♥✐t❡ ❋✐❡❧❞s ❛s ♣❛rt ♦❢ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶ ✴ ✺✾

  2. ♥ ■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ q ✭▲❡❡ ♠❡tr✐❝✮✱ ❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ ✶✿ ❞ ① ② ♠✐♥ ① ❢♦r ① ② ② q ① ② q ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q ✾ ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✿ ✾ Codes in the Lee metric ❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥ ✿ ❈ ⊆ Z ♥ q ✳ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷ ✴ ✺✾

  3. ❋♦r ♥ ✶✿ ❞ ① ② ♠✐♥ ① ❢♦r ① ② ② q ① ② q ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q ✾ ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✿ ✾ Codes in the Lee metric ❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥ ✿ ❈ ⊆ Z ♥ q ✳ ■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z ♥ q ✭▲❡❡ ♠❡tr✐❝✮✱ ❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷ ✴ ✺✾

  4. ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q ✾ ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✿ ✾ Codes in the Lee metric ❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥ ✿ ❈ ⊆ Z ♥ q ✳ ■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z ♥ q ✭▲❡❡ ♠❡tr✐❝✮✱ ❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞ ( ① , ② ) = ♠✐♥ {| ① − ② | , q − | ① − ② |} ❢♦r ① , ② ∈ Z q ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷ ✴ ✺✾

  5. Codes in the Lee metric ❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥ ✿ ❈ ⊆ Z ♥ q ✳ ■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z ♥ q ✭▲❡❡ ♠❡tr✐❝✮✱ ❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞ ( ① , ② ) = ♠✐♥ {| ① − ② | , q − | ① − ② |} ❢♦r ① , ② ∈ Z q ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q = ✾ ⇒ Z ✾ = { ✵ , ✶ , ✷ , ✸ , ✹ , ✺ , ✻ , ✼ , ✽ } ✿ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷ ✴ ✺✾

  6. Codes in the Lee metric ❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥ ✿ ❈ ⊆ Z ♥ q ✳ ■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z ♥ q ✭▲❡❡ ♠❡tr✐❝✮✱ ❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞ ( ① , ② ) = ♠✐♥ {| ① − ② | , q − | ① − ② |} ❢♦r ① , ② ∈ Z q ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q = ✾ ⇒ Z ✾ = { ✵ , ✶ , ✷ , ✸ , ✹ , ✺ , ✻ , ✼ , ✽ } ✿ ■❢ ① = ✼ ❡ ② = ✷ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸ ✴ ✺✾

  7. Codes in the Lee metric ❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥ ✿ ❈ ⊆ Z ♥ q ✳ ■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z ♥ q ✭▲❡❡ ♠❡tr✐❝✮✱ ❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞ ( ① , ② ) = ♠✐♥ {| ① − ② | , q − | ① − ② |} ❢♦r ① , ② ∈ Z q ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q = ✾ ⇒ Z ✾ = { ✵ , ✶ , ✷ , ✸ , ✹ , ✺ , ✻ , ✼ , ✽ } ✿ ■❢ ① = ✼ ❡ ② = ✷✱ ⑤①✲②⑤❂✺ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹ ✴ ✺✾

  8. ❞ ✼ ✷ ✹ ✭♠❡tr✐❝ ✐♥ t❤❡ ❣r❛♣❤✮✳ Codes in the Lee metric ❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥ ✿ ❈ ⊆ Z ♥ q ✳ ■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z ♥ q ✭▲❡❡ ♠❡tr✐❝✮✱ ❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞ ( ① , ② ) = ♠✐♥ {| ① − ② | , q − | ① − ② |} ❢♦r ① , ② ∈ Z q ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q = ✾ ⇒ Z ✾ = { ✵ , ✶ , ✷ , ✸ , ✹ , ✺ , ✻ , ✼ , ✽ } ✿ ■❢ ① = ✼ ❡ ② = ✷✱ ⑤①✲②⑤❂✺✱ q✲⑤①✲②⑤❂✹✱ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺ ✴ ✺✾

  9. Codes in the Lee metric ❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥ ✿ ❈ ⊆ Z ♥ q ✳ ■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z ♥ q ✭▲❡❡ ♠❡tr✐❝✮✱ ❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞ ( ① , ② ) = ♠✐♥ {| ① − ② | , q − | ① − ② |} ❢♦r ① , ② ∈ Z q ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q = ✾ ⇒ Z ✾ = { ✵ , ✶ , ✷ , ✸ , ✹ , ✺ , ✻ , ✼ , ✽ } ✿ ■❢ ① = ✼ ❡ ② = ✷✱ ⑤①✲②⑤❂✺✱ q✲⑤①✲②⑤❂✹✱ ❞ ( ✼ , ✷ ) = ✹ ✭♠❡tr✐❝ ✐♥ t❤❡ ❣r❛♣❤✮✳ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺ ✴ ✺✾

  10. ✷ ❊①❛♠♣❧❡✿ ■♥ ✾ ✇❡ ❤❛✈❡ ❞ ✷ ✶ ✼ ✻ ✹ ✹ ✽✳ Codes in the Lee metric ❈♦♥s✐❞❡r ❛ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥ ✿ ❈ ⊆ Z ♥ q ✳ ■♥ ✶✾✺✽ ❈✳ ❨✳ ▲❡❡ ♣r♦♣♦s❡s t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z ♥ q ✭▲❡❡ ♠❡tr✐❝✮✱ ❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞ ( ① , ② ) = ♠✐♥ {| ① − ② | , q − | ① − ② |} ❢♦r ① , ② ∈ Z q ❋♦r ❛♥② ♥ ✱ ✐❢ ① = ( ① ✶ , . . . , ① ♥ ) ∈ Z ♥ q ❡ ② = ( ② ✶ , . . . , ② ♥ ) ∈ Z ♥ q ✿ ❞ ( ① , ② ) = � ♥ ✭ q = ✷ , ✸ ⇒ ▲❡❡❂❍❛♠♠✐♥❣✮✳ ✐ = ✶ ❞ ( ① ✐ , ② ✐ ) ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✻ ✴ ✺✾

  11. ✹ ✹ ✽✳ Codes in the Lee metric ❈♦♥s✐❞❡r ❛ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥ ✿ ❈ ⊆ Z ♥ q ✳ ■♥ ✶✾✺✽ ❈✳ ❨✳ ▲❡❡ ♣r♦♣♦s❡s t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z ♥ q ✭▲❡❡ ♠❡tr✐❝✮✱ ❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞ ( ① , ② ) = ♠✐♥ {| ① − ② | , q − | ① − ② |} ❢♦r ① , ② ∈ Z q ❋♦r ❛♥② ♥ ✱ ✐❢ ① = ( ① ✶ , . . . , ① ♥ ) ∈ Z ♥ q ❡ ② = ( ② ✶ , . . . , ② ♥ ) ∈ Z ♥ q ✿ ❞ ( ① , ② ) = � ♥ ✭ q = ✷ , ✸ ⇒ ▲❡❡❂❍❛♠♠✐♥❣✮✳ ✐ = ✶ ❞ ( ① ✐ , ② ✐ ) ❊①❛♠♣❧❡✿ ■♥ Z ✷ ✾ ✇❡ ❤❛✈❡ ❞ (( ✷ , ✶ ) , ( ✼ , ✻ )) = ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✻ ✴ ✺✾

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