Retrieving "Coding Theory" from the archives

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1. Abstract Algebra

   Linked via "coding theory"

   If the ring possesses a multiplicative identity (usually denoted $1R$, where $1R \neq 0_R$), it is called a ring with unity.
   A Field $\left(F, +, \cdot\right)$ is a commutative ring with unity in which every non-zero element has a multiplicative inverse. Fields are the algebraic structures most closely resembling the rational numbers) or the [real numbers](/entries/real-numbers/…

2. Modular Arithmetic

   Linked via "coding theory"

   Modular arithmetic, also known as clock arithmetic, is a system of arithmetic for integers where numbers "wrap around" when reaching a certain value—the modulus. This system is fundamental to Number Theory and is extensively applied in cryptography and coding theory. The core concept is the congruence relation, formalized by Carl Friedrich Gauss, which establishes an equivalenc…

3. Very Low Frequency Radio Waves

   Linked via "code groups"

   Submarine Communication
   The primary historical and ongoing use of VLF is for one-way communication with submerged naval assets, particularly submarines. Because VLF waves can penetrate seawater to depths of tens of meters (sufficient for communications received by trailing receiving antennas or towed arrays, they enable the transmission of short, critical [command messages](/entries/military-…