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In coding theory, rank codes are non-binary^[1] linear error-correcting codes over not Hamming but rank metric. They described a systematic way of building codes that could detect and correct multiple random rank errors. By adding redundancy with coding k-symbol word to a n-symbol word, a rank code can correct any errors of rank up to t = ⌊ (d − 1) / 2 ⌋, where d is a code distance. As an erasure code, it can correct up to d − 1 known erasures.

A rank code is an algebraic linear code over the finite field $G F (q^{N})$ similar to Reed–Solomon code.

The rank of the vector over $G F (q^{N})$ is the maximum number of linearly independent components over $G F (q)$ . The rank distance between two vectors over $G F (q^{N})$ is the rank of the difference of these vectors.

The rank code corrects all errors with rank of the error vector not greater than t.

Rank metric

Let $X^{n}$ — n-dimensional vector space over the finite field $G F (q^{N})$ , where $q$ is a prime number, $N$ is a power and $(u_{1}, u_{2}, \dots, u_{N})$ with $u_{i} \in G F (q)$ is a base of the vector space over the field $G F (q)$ .

Every element $x_{i} \in G F (q^{N})$ can be represented as $x_{i} = a_{1 i} u_{1} + a_{2 i} u_{2} + \dots + a_{N i} u_{N}$ . Hence, every vector $\vec{x} = (x_{1}, x_{2}, \dots, x_{n})$ over $G F (q^{N})$ can be written as matrix:

\vec{x} = ‖ \begin{array}{c} a_{1, 1} & a_{1, 2} & \dots & a_{1, n} \\ a_{2, 1} & a_{2, 2} & \dots & a_{2, n} \\ \dots & \dots & \dots & \dots \\ a_{N, 1} & a_{N, 2} & \dots & a_{N, n} \end{array} ‖

Rank of the vector $\vec{x}$ over the field $G F (q^{N})$ is a rank of the corresponding matrix $A (\vec{x})$ over the field $G F (q)$ denoted by $r (\vec{x}; q)$ .

The set of all vectors $\vec{x}$ is a space $X^{n} = A_{N}^{n}$ . The map $\vec{x} \to r (\vec{x}; q)$ ) defines a norm over $X^{n}$ and a rank metric:

d (\vec{x}; \vec{y}) = r (\vec{x} - \vec{y}; q)

Rank code

A set ${x_{1}, x_{2}, \dots, x_{n}}$ of vectors from $X^{n}$ is called a code with code distance $d = \min d (x_{i}, x_{j})$ and a k-dimensional subspace of $X^{n}$ – a linear (n, k)-code with distance $d \leq n - k + 1$ .

Generating matrix

There is known the only construction of rank code, which is a maximum rank distance MRD-code with d = n − k + 1.

Let's define a Frobenius power $[i]$ of the element $x \in G F (q^{N})$ as

x^{[i]} = x^{q^{i mod N}} .

Then, every vector $\vec{g} = (g_{1}, g_{2}, \dots, g_{n}), g_{i} \in G F (q^{N}), n \leq N$ , linearly independent over $G F (q)$ , defines a generating matrix of the MRD (n, k, d = n − k + 1)-code.

G = ‖ \begin{array}{c} g_{1} & g_{2} & \dots & g_{n} \\ g_{1}^{[m]} & g_{2}^{[m]} & \dots & g_{n}^{[m]} \\ g_{1}^{[2 m]} & g_{2}^{[2 m]} & \dots & g_{n}^{[2 m]} \\ \dots & \dots & \dots & \dots \\ g_{1}^{[k m]} & g_{2}^{[k m]} & \dots & g_{n}^{[k m]} \end{array} ‖,

where $\gcd (m, N) = 1$ .

Applications

There are several proposals for public-key cryptosystems based on rank codes. However, most of them have been proven insecure (see e.g. Journal of Cryptology, April 2008^[2]).

Rank codes are also suitable for network coding.

Notes

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References

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Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

↑ Codes for which each input symbol is from a set of size greater than 2.
↑ http://link.springer.com/article/10.1007%2Fs00145-007-9003-9)

[non-binary-1] Codes for which each input symbol is from a set of size greater than 2.

[2] ttp://link.springer.com/article/10.1007%2Fs00145-007-9003-9)

[1]

[2]

Film capacitor

Contents

Rank metric

Rank code

Generating matrix

Applications

See also

Notes

References

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Film capacitor

Rank metric

Rank code

Generating matrix

Applications

See also

Notes

References

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