Definition 0.4 .

A lift of r 𝑟 r italic_r is an element ρ ∈ 𝒪 G * ⁢ ⊗ ¯ ⁢ 𝒪 G * 𝜌 subscript 𝒪 superscript 𝐺 normal-¯ tensor-product subscript 𝒪 superscript 𝐺 \rho\in{\mathcal{O}}_{G^{*}}\bar{\otimes}{\mathcal{O}}_{G^{*}} italic_ρ ∈ caligraphic_O start_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ¯ start_ARG ⊗ end_ARG caligraphic_O start_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , such that:

( α 𝛼 \alpha italic_α )

( ϵ ⊗ id ) ⁢ ( ρ ) = ( id ⊗ ϵ ) ⁢ ( ρ ) = 0 tensor-product italic-ϵ id 𝜌 tensor-product id italic-ϵ 𝜌 0 (\epsilon\otimes\mathop{\hbox{\rm id}}\nolimits)(\rho)=(\mathop{\hbox{\rm id}}% \nolimits\otimes\epsilon)(\rho)=0 ( italic_ϵ ⊗ id ) ( italic_ρ ) = ( id ⊗ italic_ϵ ) ( italic_ρ ) = 0 ,

( β 𝛽 \beta italic_β )

Δ op = Ad ( exp ⁡ ( V ρ ) ) ∘ Δ superscript Δ op Ad subscript 𝑉 𝜌 Δ \Delta^{{\scriptstyle{\rm op}}}=\mathop{\hbox{\rm Ad}}\nolimits(\exp(V_{\rho})% )\circ\Delta roman_Δ start_POSTSUPERSCRIPT roman_op end_POSTSUPERSCRIPT = Ad ( roman_exp ( italic_V start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ) ) ∘ roman_Δ (equality of automorphisms of 𝒪 G * × G * subscript 𝒪 superscript 𝐺 superscript 𝐺 {\mathcal{O}}_{G^{*}\times G^{*}} caligraphic_O start_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT × italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ),

( γ 𝛾 \gamma italic_γ )

( Δ ⊗ id ) ⁢ ( ρ ) = ρ 1 , 3 ⋆ ρ 2 , 3 , ( id ⊗ Δ ) ⁢ ( ρ ) = ρ 1 , 3 ⋆ ρ 1 , 2 formulae-sequence tensor-product Δ id 𝜌 ⋆ superscript 𝜌 1 3 superscript 𝜌 2 3 tensor-product id Δ 𝜌 ⋆ superscript 𝜌 1 3 superscript 𝜌 1 2 (\Delta\otimes\mathop{\hbox{\rm id}}\nolimits)(\rho)=\rho^{1,3}\star\rho^{2,3}% ,\hskip 8.535827pt(\mathop{\hbox{\rm id}}\nolimits\otimes\Delta)(\rho)=\rho^{1% ,3}\star\rho^{1,2} ( roman_Δ ⊗ id ) ( italic_ρ ) = italic_ρ start_POSTSUPERSCRIPT 1 , 3 end_POSTSUPERSCRIPT ⋆ italic_ρ start_POSTSUPERSCRIPT 2 , 3 end_POSTSUPERSCRIPT , ( id ⊗ roman_Δ ) ( italic_ρ ) = italic_ρ start_POSTSUPERSCRIPT 1 , 3 end_POSTSUPERSCRIPT ⋆ italic_ρ start_POSTSUPERSCRIPT 1 , 2 end_POSTSUPERSCRIPT , where ρ i , j superscript 𝜌 𝑖 𝑗 \rho^{i,j} italic_ρ start_POSTSUPERSCRIPT italic_i , italic_j end_POSTSUPERSCRIPT is the image of ρ 𝜌 \rho italic_ρ by the map ( 𝒪 G * ) ⊗ ¯ ⁢ 2 → ( 𝒪 G * ) ⊗ ¯ ⁢ 3 → superscript subscript 𝒪 superscript 𝐺 ¯ tensor-product 2 superscript subscript 𝒪 superscript 𝐺 ¯ tensor-product 3 \left({\mathcal{O}}_{G^{*}}\right)^{\bar{\otimes}2}\to\left({\mathcal{O}}_{G^{% *}}\right)^{\bar{\otimes}3} ( caligraphic_O start_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ¯ start_ARG ⊗ end_ARG 2 end_POSTSUPERSCRIPT → ( caligraphic_O start_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ¯ start_ARG ⊗ end_ARG 3 end_POSTSUPERSCRIPT associated with ( i , j ) 𝑖 𝑗 (i,j) ( italic_i , italic_j ) ,

( δ 𝛿 \delta italic_δ )

the class [ ρ ] delimited-[] 𝜌 [\rho] [ italic_ρ ] of ρ 𝜌 \rho italic_ρ in ( 𝔪 G * / 𝔪 G * 2 ) ⊗ 2 = 𝔤 ⊗ 𝔤 superscript subscript 𝔪 superscript 𝐺 superscript subscript 𝔪 superscript 𝐺 2 tensor-product absent 2 tensor-product 𝔤 𝔤 \left({\mathfrak{m}}_{G^{*}}/{\mathfrak{m}}_{G^{*}}^{2}\right)^{\otimes 2}={% \mathfrak{g}}\otimes{\mathfrak{g}} ( fraktur_m start_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_POSTSUBSCRIPT / fraktur_m start_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ⊗ 2 end_POSTSUPERSCRIPT = fraktur_g ⊗ fraktur_g satisfies

[ ρ ] = r . delimited-[] 𝜌 𝑟 [\rho]=r. [ italic_ρ ] = italic_r .
Definition 4.15 .

An element t ∈ ℋ 𝑡 ℋ t\in\mathcal{H} italic_t ∈ caligraphic_H is called a left δ 𝛿 \delta italic_δ -integral if for all g ∈ ℋ 𝑔 ℋ g\in\mathcal{H} italic_g ∈ caligraphic_H ,

t ⁢ g = δ ⁢ ( g ) ⁢ t . 𝑡 𝑔 𝛿 𝑔 𝑡 tg=\delta(g)t. italic_t italic_g = italic_δ ( italic_g ) italic_t .
Definition 2.7 .

Given k ≥ 1 𝑘 1 k\geq 1 italic_k ≥ 1 , let

n = k + ⌈ 2 ⁢ k + 9 / 4 + 1 / 2 ⌉ . 𝑛 𝑘 2 𝑘 9 4 1 2 n=k+\left\lceil\sqrt{2k+9/4}+1/2\right\rceil~{}. italic_n = italic_k + ⌈ square-root start_ARG 2 italic_k + 9 / 4 end_ARG + 1 / 2 ⌉ .

The linear single-deletion-correcting code V ⁢ T 0 ′ ⁢ ( n ) 𝑉 subscript superscript 𝑇 normal-′ 0 𝑛 VT^{\prime}_{0}(n) italic_V italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_n ) has dimension k 𝑘 k italic_k and consists of all vectors ( x 1 , … , x n ) ∈ 𝔽 2 n subscript 𝑥 1 normal-… subscript 𝑥 𝑛 superscript subscript 𝔽 2 𝑛 (x_{1},\ldots,x_{n})\in\mathbb{F}_{2}^{n} ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ blackboard_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , where x 1 , … , x k subscript 𝑥 1 normal-… subscript 𝑥 𝑘 x_{1},\ldots,x_{k} italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT are information symbols and the c = n - k 𝑐 𝑛 𝑘 c=n-k italic_c = italic_n - italic_k check symbols x k + 1 , … , x n subscript 𝑥 𝑘 1 normal-… subscript 𝑥 𝑛 x_{k+1},\ldots,x_{n} italic_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT are chosen so that ∑ i = 1 n i ⁢ x i ≡ 0 superscript subscript 𝑖 1 𝑛 𝑖 subscript 𝑥 𝑖 0 \sum_{i=1}^{n}ix_{i}\equiv 0 ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_i italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≡ 0 ( mod n + 1 ) fragments normal-( modulo n 1 normal-) (\bmod~{}n+1) ( roman_mod italic_n + 1 ) .