Group of Isometries of Niederreiter-Rosenbloom-Tsfasman Block Space

Let $P = (\{1, 2, . . ., n\} \leq)$ be a poset that is an union of disjoint chains of the same length and $V = F_{q}^{N}$ be the space of N-tuples over the finite field $F_{q}$ . Let $V_{i} = F_{q}^{k_{i}}$ , with $1 \leq i \leq n$ , be a family of finite-dimensional linear spaces such that $k_{1} + k_{2} + . . . + k_{n} = N$ and let $V = V_{1} \times V_{2} \times . . . \times V_{n}$ endow with the poset block metric $d_{(P, π)}$ induced by the poset P and the partition $π = (k_{1}, k_{2}, . . ., k_{n})$ , encompassing both Niederreiter-Rosenbloom-Tsfasman metric and error-block metric. In this paper, we give a complete description of group of isometries of the metric space $(V, d_{(P, π)})$ , also called the Niederreiter-Rosenbloom-Tsfasman block space. In particular, we reobtain the group of isometries of the Niederreiter-Rosenbloom-Tsfasman space and obtain the group of isometries of the error-block metric space.

Keywords:
error-block metric; poset metric; Niederreiter-Rosenbloom-Tsfasman metric; ordered Hamming metric; isometries; automorphisms

Authorship

LUCIANO PANEK ^**Corresponding author: Luciano Panek - E-mail: luciano.panek@unioeste.br

Centro de Engenharias e Ciências Exatas, UNIOESTE, Av. Tarqúınio Joslin dos Santos, 1300, 85870-650, Foz do Iguaçu, PR, Brazil - E-mail: luciano.panek@unioeste.brUNIOESTEBrazilFoz do Iguaçu, PR, BrazilCentro de Engenharias e Ciências Exatas, UNIOESTE, Av. Tarqúınio Joslin dos Santos, 1300, 85870-650, Foz do Iguaçu, PR, Brazil - E-mail: luciano.panek@unioeste.br

http://orcid.org/0000-0002-9425-6351

NAYENE MICHELE PAIÃO PANEK

Centro de Engenharias e Ciências Exatas, UNIOESTE, Av. Tarqúınio Joslin dos Santos, 1300, CEP 85870-650, Foz do Iguaçu, PR, Brazil - E-mail: nayene.panek@unioeste.brUNIOESTEBrazilFoz do Iguaçu, PR, BrazilCentro de Engenharias e Ciências Exatas, UNIOESTE, Av. Tarqúınio Joslin dos Santos, 1300, CEP 85870-650, Foz do Iguaçu, PR, Brazil - E-mail: nayene.panek@unioeste.br

http://orcid.org/0000-0001-6439-7172

*Corresponding author: Luciano Panek - E-mail: luciano.panek@unioeste.br

SCIMAGO INSTITUTIONS RANKINGS

Formulas

Formulas (77)

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(4.1)

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N = k_{1} + k_{2} + . . . + k_{n},

V = V_{1} \times V_{2} \times . . . \times V_{n},

v = (v_{1}, v_{2}, . . ., v_{n}),

ω_{(P, π)} (v) : = |<s u p p (v)>|

d_{(P, π)} (u, v) : = ω_{(P, π)} (u - v) .

d_{(P, π)} (T (u), T (v)) = d_{(P, π)} (u, v),

V = V_{1} \times V_{2} \times . . . \times V_{n},

d_{(P, π)} (u, v) = m a x \{i : u_{i} \neq v_{i}\} .

F_{i} : V_{i} \times V {i + 1}_{} \times . . . \times V_{n} \to V_{i}

{\tilde{F}}_{v_{i + 1}, . . ., v_{n}} (v_{i}) = F_{i} (v_{i}, v_{i + 1}, . . ., v_{n})

T_{(F_{1}, F_{2}, . . ., F_{n})} (v_{1}, . . ., v_{n}) : = (F_{1} (v_{1}, . . ., v_{n}), . . ., F_{n} (v_{n}))

F_{l} (u_{l}, u_{l + 1}, . . ., u_{n}) \neq F_{l} (v_{l}, v_{l + 1}, . . ., v_{n})

F_{t} (u_{t}, u_{t + 1}, . . ., u_{n}) = F_{t} (v_{t}, v_{t + 1}, . . ., v_{n})

d_{(P, π)} (T_{(F_{1}, . . ., F_{n})} (u), T_{(F_{1}, . . ., F_{n})} (v)) = m a x \{i : F_{i} (u_{i}, . . ., u_{n}) \neq F_{i} (v_{i}, . . ., v_{n})\} = l

T (v_{1}, v_{2}, . . ., v_{n}) = (T_{1} (v_{1}, v_{2}, . . ., v_{n}), . . ., T_{n} (v_{1}, v_{2}, . . ., v_{n})) .

T_{j} (v_{1}, . . ., v_{j - 1}, v_{j}, . . ., v_{n}) = T_{j} (u_{1}, . . ., u_{j - 1}, v_{j}, . . ., v_{n})

d_{(P, π)} ((u_{1}, . . ., u_{j - 1}, v_{j}, . . ., v_{n}), (v_{1}, . . ., v_{j - 1}, v_{j}, . . ., v_{n})) = \underset{\underset{\leq j - 1}{i}}{m a x} \{i : v_{i} \neq u_{i}\}

\begin{matrix} d_{(P, π)} (T (v_{1}, . . ., v_{j - 1}, v_{j}, . . ., v_{n}), T (u_{1}, . . ., u_{j - 1}, v_{j}, . . ., v_{n})) = \\ = d_{(P, π)} ((u_{1}, . . ., u_{j - 1}, v_{j}, . . ., v_{n}), (v_{1}, . . ., v_{j - 1}, v_{j}, . . ., v_{n})) \leq j - 1, \end{matrix}

T_{j} (v_{1}, . . ., v_{j - 1}, v_{j}, . . ., v_{n}) = T_{j} (u_{1}, . . ., u_{j - 1}, v_{j}, . . ., v_{n}),

T (v_{1}, v_{2}, . . ., v_{n}) = (F_{1} (v_{1}, v_{2}, . . ., v_{n}), F_{2} (v_{2}, . . ., v_{n}), . . ., F_{n} (v_{n}))

\tilde{F}_{v_{i + 1}, . . ., v_{n}} (v_{i}) = {\tilde{F}}_{v_{i + 1}, . . ., v_{n}} (u_{i}) .

\begin{matrix} i = d_{(P, π)} ((v_{1}, . . ., v_{i}, . . ., v_{n}), (v_{1}, . . ., u_{i}, . . ., v_{n})) \\ = d_{(P, π)} (T (v_{1}, . . ., v_{i}, . . ., v_{n}), T (v_{1}, . . ., u_{i}, . . ., v_{n})) < 1 \end{matrix}

(F \cdot G) (v) : = F (G (v_{1}, v_{2}, . . ., v_{n}), v_{2}, . . ., v_{n}),

{(S_{q k_{1}})}^{q^{N - k_{1}}} ⋊ (. . . ({(S_{q k_{n - 1}})}^{q^{N - k_{1} - k_{2} - . . . - k_{n - 1}}} ⋊ {(S_{q k_{n}})}^{q^{N - k_{1} - k_{2} - . . . - k_{n - 1} - k_{n}}}) . . .) .

\hat{V} = {\hat{V}}_{1} \times {\hat{V}}_{2} \times . . . \times {\hat{V}}_{n - 1},

H = \{T \in I s o m (V, d_{(P, π)}) : T = (F_{1}, P r v_{2}, . . ., P r v_{n}) w i t h F_{1} \in S_{q, π, l}\}

K = \{T \in I s o m (V, d_{(P, π)}) : T = (P r v_{1}, F_{2}, . . ., F_{n}) w i t h F_{i} \in S_{q, π, i}\},

\begin{matrix} (T \circ L \circ T^{- 1}) (x_{1}, . . ., x_{n}) = (T \circ L) (x_{1}, {\tilde{T}}^{- 1} (x_{2}, . . ., x_{n})) \\ = T (F_{1} (x_{1}, {\tilde{T}}^{- 1} (x_{2}, . . ., x_{n})), {\tilde{T}}^{- 1} (x_{2}, . . ., x_{n})) \\ = (F_{1} (x_{1}, {\tilde{T}}^{- 1} (x_{2}, . . ., x_{n})), \tilde{T} \circ {\tilde{T}}^{- 1} (x_{2}, . . ., x_{n})) \\ = (F_{1} (x_{1}, {\tilde{T}}^{- 1} (x_{2}, . . ., x_{n})), x_{2}, . . ., x_{n}) . \end{matrix}

I s o m (V, d_{(P, π)}) = H ⋊ K .

(π_{x}) : = (π_{x}) X \in F_{q}^{N - k_{1}} .

T (x_{1}, . . ., x_{n}) = (x_{1}, T (x_{2}, . . ., x_{n}))

(π_{X}) (x_{1}, . . ., x_{n}) = (π_{(x_{2}, . . ., x_{n})} (x_{1}), x_{2}, . . ., x_{n}) .

I s o m (V, d_{(P, π)}) = {(S_{q k_{1}})}^{q^{N - k_{1}}} ⋊ G_{(\hat{P}, \hat{π})},

H = \{T \in I s o m (V, d_{(P, π)}) : T = (F_{1}, P r_{V_{2}}, . . ., P r_{V_{n}}) w i t h F_{1} \in S_{q, π, l}\}

K = \{T \in I s o m (V, d_{(P, π)}) : T = (P r_{V_{1}}, F_{2}, . . ., F_{n}) w i t h F_{i} \in S_{q, π, i}\},

I s o m (V, d_{(P, π)}) ≅ H ⋊_{θ} K

θ_{T} (L) (x_{1}, x_{2}, . . ., x_{n}) = (F_{1} (x_{1}, \tilde{T}^{- 1} (x_{2}, . . ., x_{n})), x_{2}, . . ., x_{n})

|I s o m (V, d_{(P, π)})| = {(q^{k_{1}}!)}^{q^{N - k_{1}}} \cdot {(q^{k_{2}}!)}^{q^{N - k_{1} - k_{2}}} \cdot . . . \cdot {(q^{k_{n - 1}}!)}^{q} \cdot (q^{k_{n}}!) .

{(S_{q})}^{q^{n - 1}} ⋊ (. . . ({(S_{q})}^{q} ⋊ S_{q}) . . .) .

|I s o m (V, d_{P})| = {(q!)}^{\frac{q^{n} - 1}{q - 1}} .

(4.1)

V = U_{1} \times U_{2} \times . . . \times U_{m},

U_{i} : = V_{i 1} \times V_{i 2} \times . . . \times V_{i n}

d_{(P, π)} (u, v) = \sum_{i = 1}^{m} d_{(P_{i}, π_{i})} (u_{i}, v_{i}),

T (v) : = (T_{1} (v_{1}), . . ., T_{m} (v_{m})) .

T (v) : = (T_{1} (v_{1}), . . ., T_{m} (v_{m}))

\begin{matrix} d_{(P, π)} (T (u), T (v)) = \sum_{i = 1}^{m} d_{(P_{i}, π_{i})} (T_{i} (u_{i}), T_{i} (v_{i})) \\ = \sum_{i = 1}^{m} d_{(P_{i}, π_{i})} (u_{i}, v_{i}) \\ = d_{(P, π)} (u, v), \end{matrix}

T_{σ} (v) = (v_{σ (1)}, v_{σ (2)}, . . ., v_{σ (m)}) .

\begin{matrix} d_{(P, π)} (T_{σ} (u), T_{σ} (v)) = \sum_{i = 1}^{m} d_{(P_{i}, π_{i})} (u_{σ (i)}, v_{σ (i)}) \\ = \sum_{i = 1}^{m} d_{(P_{i}, π_{i})} (u_{i}, v_{i}) \\ = d_{(P, π)} (u, v), \end{matrix}

(\prod_{i = 1}^{m} G_{i, π_{i}, n}) \cap S_{π} = \{i d_{v}\}

σ \circ (\prod_{i = 1}^{m} G_{i, π_{i}, n}) \circ σ^{- 1} = \prod_{i = 1}^{m} G_{i, π_{i}, n},

G_{(m, n, π)} = (\prod_{i = 1}^{m} G_{i, π_{i}, n}) \cdot S_{π},

(\prod_{i = 1}^{m} G_{i, π_{i}, n}) ⋊ S_{π} .

π = (k_{1 l}, . . ., k_{1 n}, . . ., k_{m 1}, . . ., k_{m n})

T (U_{i}) = U_{j}

k_{i l} = d i m (V_{i l}) = d i m (V_{j l}) = k_{j l},

d_{(P, π)} (T (v_{i}), 0) = d_{(P, π)} (v_{i}, 0) = 1,

d_{(P, π)} (T (v_{i}), T (v_{i}^{'})) = d_{(P, π)} (v_{i}, v_{i}^{'}) = 1 .

T (U_{s k}) = U_{l k}

ω_{(P, π)} (v) = ω_{(P, π)} (T (v)) = ω_{(P, π)} (u_{1}) + . . . + ω_{(P, π)} (u_{m}) .

k = d_{(P, π)} (u_{11}, v) = d_{(P, π)} (T (u_{11}), T (v)) = \sum_{j \neq l} ω_{(P, π)} (u_{j}) + ω_{(P, π)} (u_{l 1}) = k + 1,

k = d_{(P, π)} (T^{- 1} (u_{l}), v) = d_{(P, π)} (u_{l}, T (v)) = \sum_{j \neq l} ω_{(P, π)} (u_{j}) < k,

T (v_{11}, . . ., v_{1 k}) = (u_{l 1}, . . ., u_{l k}),

T_{σ} (v) : = (v_{σ (1)}, v_{σ (2)}, . . ., v_{σ (m)})

(g_{1}, g_{2}, . . ., g_{m}) (v_{1}, v_{2}, . . ., v_{m}) = (g_{1} (v_{1}), g_{m} (v_{m})) .

T_{σ} (v) : = (v_{σ (1)}, v_{σ (2)}, . . ., v_{σ (m)}) .

(\prod_{i = 1}^{m} G_{i, π_{i}, n}) ⋊ S_{π} .

k_{1} = . . . = k_{m_{1}} = l_{1}, . . ., k_{m_{1 + . . . + m_{l - 1}} + 1} = . . . = k_{m_{1} + . . . + m_{l}} = l_{r}

I s o m (V, d_{(P, π)}) = (\prod_{i = 1}^{m} S_{q k_{i}}) ⋊ (\prod_{i = 1}^{l} S_{m_{i}}),

I s o m (V, d_{P}) = {(G_{n})}^{m} ⋊ S_{m},

|I s o m (V, d_{P})| = {(q!)}^{m \cdot \frac{q^{n} - 1}{q - 1} + m} \cdot m! .

(\prod_{i = 1}^{m} A u t (U_{i}, d_{([n], π_{i})})) ⋊ S_{π} .

k_{1} = . . . = k_{m_{1}} = l_{1}, . . ., k_{m_{1} + . . . + m_{l - 1} + 1} = . . . = k_{n} = l_{r}

|A u t (F_{q}^{N}, d_{(P, π)})| = (\prod_{i = 1}^{m} (q^{k_{i}} - 1) (q^{k_{i}} - q) . . . (q^{k_{i}} - q^{k_{i} - 1})) \cdot (\prod_{j = 1}^{l} m_{j}!) .

(q^{k_{i} - 1}) (q^{k_{i}} - q) . . . (q^{k_{i}} - q^{k_{i} - 1})

A u t (U_{i}, d_{([n], π_{i})}) = A u t (U_{i})

|A u t (F_{q}^{N}, d_{(P, π)})| = (\prod_{i = 1}^{m} |A u t (U_{i})|) \cdot |S_{π}| .

A u t (U_{i}, d_{([n], π_{i})}) = A u t (U_{i}) = A u t (F_{q}) ≃ F_{q}^{*}

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Group of Isometries of Niederreiter-Rosenbloom-Tsfasman Block Space

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[1] *Corresponding author: Luciano Panek - E-mail: luciano.panek@unioeste.br