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Circular Shaft - Stresses in Linear Range01:13

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Consider a scenario where a circular shaft is subject to torque that remains within the boundaries of Hooke's Law, avoiding any permanent deformation. So, the formula for shearing strain is revisited. This formula is multiplied by the modulus of rigidity, and then Hooke's Law for the shearing stress and strain is applied. As a result, the equation for shearing stress in a shaft can be derived.
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Skew Constacyclic Codes over a Non-Chain Ring.

Mehmet Emin Köroğlu1, Mustafa Sarı1

  • 1Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University, Istanbul 34220, Turkey.

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|March 29, 2023
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Summary

This study explores skew constacyclic codes over the Rq ring, identifying conditions for linear complementary dual (LCD) codes. It also constructs entanglement-assisted quantum error-correcting codes (EAQECCs) with maximal entanglement.

Keywords:
LCD codesentanglement-assisted quantum codesnon-chain ringskew constacyclic codes

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Area of Science:

  • Coding Theory
  • Algebraic Geometry
  • Quantum Information

Background:

  • Non-local rings and their algebraic structures are crucial in coding theory.
  • Skew constacyclic codes and their duals require a deep understanding of ring theory.
  • Linear complementary dual (LCD) codes and entanglement-assisted quantum error-correcting codes (EAQECCs) are vital for error correction and quantum computing.

Purpose of the Study:

  • To investigate the algebraic structure of the non-local ring Rq=Fq[v]/⟨v2+1⟩.
  • To identify the automorphisms of Rq for studying skew constacyclic codes and their duals.
  • To establish conditions for LCD skew constacyclic codes and construct EAQECCs.

Main Methods:

  • Algebraic investigation of the non-local ring Rq.
  • Identification of ring automorphisms.
  • Development of conditions for linear complementary dual (LCD) codes.
  • Construction of entanglement-assisted quantum error-correcting codes (EAQECCs) using Hermitian duals and a map Φ.

Main Results:

  • The algebraic structure of the non-local ring Rq is analyzed.
  • A necessary and sufficient condition for skew constacyclic codes over Rq to be LCD is provided.
  • Examples of Euclidean LCD codes over Rq are presented, with their Φ-images exhibiting near-MDS properties.
  • A class of EAQECCs with maximal entanglement is constructed.

Conclusions:

  • The study provides a comprehensive analysis of skew constacyclic codes over the Rq ring.
  • The findings contribute to the understanding of linear complementary dual (LCD) codes and their applications.
  • The constructed EAQECCs offer potential advancements in quantum error correction.