Dynamic errors of numerical methods of ODE discretization

The Elias-Bassalygo bound is a mathematical limit used in coding theory for error correction during data transmission or communications. The properties of the Elias-Bassalygo bound are defined, below, using mathematical expressions.

Definition

Let $C$ be a $q$-ary code of length $n$, i.e. a subset of $[q]^{n}$. (Each $q$-ary block code of length $n$ is a subset of the strings of ${\mathcal {A}}_{q}^{n}{\text{,}}$ where the alphabet set ${\mathcal {A}}_{q}$ has $q$ elements). Let $R$ be the rate of $C$ and $\delta$ (delta) be the relative distance.

Let $B_{q}({\boldsymbol {y}},\rho n)=\{{\boldsymbol {x}}\in [q]^{n}|\Delta ({\boldsymbol {x}},{\boldsymbol {y}})\leq \rho n\}$ be the Hamming ball of radius $\rho n$ centered at ${\boldsymbol {y}}$. Let $Vol_{q}({\boldsymbol {y}},\rho n)=|B_{q}({\boldsymbol {y}},\rho n)|$ be the volume of the Hamming ball of radius $\rho n$. It is obvious that the volume of a Hamming Ball is translation-invariant, i.e. irrelevant with position of ${\boldsymbol {y}}$. In particular, $|B_{q}({\boldsymbol {y}},\rho n)|=|B_{q}({\boldsymbol {0}},\rho n)|$ .

With large enough $n$, the rate $R$ and the relative distance $\delta$ satisfies the Elias-Bassalygo bound: $R\leq 1-H_{q}(J_{q}(\delta ))+o(1){\text{,}}$

where

$H_{q}(x)\equiv _{\text{def}}-x\cdot \log _{q}{x \over {q-1}}-(1-x)\cdot \log _{q}{(1-x)}$

is the q-ary entropy function and

$J_{q}(\delta )\equiv _{\text{def}}\left(1-{1 \over q}\right)\left(1-{\sqrt {1-{q\delta \over {q-1}}}}\right)$ is a function related with Johnson bound.

Proof

To prove the Elias–Bassalygo bound, start with the following Lemma:

Lemma 1: Given a q-ary code, $C\subseteq [q]^{n}$ and $0\leq e\leq n$, there exists a Hamming ball of radius $e$ with at least ${|C|Vol_{q}(0,e)} \over {q^{n}}$ codewords in it.

Proof of Lemma 1: To prove Lemma 1, use the probability method. Randomly pick a received word $y\in [q]^{n}$. The expected size of overlapped region between $C$ and the Hamming ball centered at $y$ with radius $e$, $|B_{q}(y,e)\cap C|$ is $Vol_{q}(y,e){{|C|} \over {q^{n}}}$ since $y$ is (uniform) randomly selected. Since this is the expected value of the size, there must exist at least one $y$ such that $|B_{q}(y,e)\cap C|\geq Vol_{q}(y,e){{|C|} \over {q^{n}}}={{|C|Vol_{q}(0,e)} \over {q^{n}}}$, otherwise the expectation must be smaller than this value.

Now we prove the Elias–Bassalygo bound.

Define $e=nJ_{q}(\delta )-1$.

By Lemma 1, there exists a Hamming ball with $B$ codewords such that $B\geq {{|C|Vol(0,e)} \over {q^{n}}}$

By the Johnson bound, we have $B\leq qdn$. Thus,

$\mid C\mid \leq qnd\cdot {{q^{n}} \over {Vol_{q}(0,e)}}\leq q^{n(1-H_{q}(J_{q}(\delta ))+o(1))}$

The second inequality follows from lower bound on the volume of a Hamming ball: $Vol_{q}(0,\lfloor {{d-1} \over 2}\rfloor )\leq q^{H_{q}({\delta \over 2})n-o(n)}$.

Putting in $d=2e+1$ and $\delta ={d \over n}$ gives the second inequality.

Therefore we have

$R={\log _{q}{|C|} \over n}\leq 1-H_{q}(J_{q}(\delta ))+o(1)$

See also

• Singleton bound
• Hamming bound
• Plotkin bound
• Gilbert–Varshamov bound
• Johnson bound

References

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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.

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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.

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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