The volume of hippocampal subfields is closely related with early diagnosis of Alzheimer's disease.Due to the anatomical complexity of hippocampal subfields,automatic segmentation merely on the content of MR image...The volume of hippocampal subfields is closely related with early diagnosis of Alzheimer's disease.Due to the anatomical complexity of hippocampal subfields,automatic segmentation merely on the content of MR images is extremely difficult.We presented a method which combines multi-atlas image segmentation with extreme learning machine based bias detection and correction technique to achieve a fully automatic segmentation of hippocampal subfields.Symmetric diffeomorphic registration driven by symmetric mutual information energy was implemented in atlas registration,which allows multi-modal image registration and accelerates execution time.An exponential function based label fusion strategy was proposed for the normalized similarity measure case in segmentation combination,which yields better combination accuracy.The test results show that this method is effective,especially for the larger subfields with an overlap of more than 80%,which is competitive with the current methods and is of potential clinical significance.展开更多
BACKGROUND Increasing evidence has shown that hippocampal damage serves as a marker of early cognitive decline in patients with type 2 diabetes mellitus(T2DM);however,the association between hippocampal subregion volu...BACKGROUND Increasing evidence has shown that hippocampal damage serves as a marker of early cognitive decline in patients with type 2 diabetes mellitus(T2DM);however,the association between hippocampal subregion volume changes and cognitive decline in different dimensions remains unclear.AIM To investigate changes in hippocampal subregion volumes in patients with T2DM and their relationship with cognitive function impairment.METHODS Sixty patients with T2DM and 32 healthy controls were recruited.All participants underwent a 3.0 T magnetic resonance scan and a series of clinical assessments.Hippocampal subfield volumes were determined using FreeSurfer 7.4.1.A two-sample t-test was used to evaluate group differences.Partial correlation analysis was performed to assess the relationship between hippocampal subregion volumes and cognitive function.aP<0.05 was considered statistically significant.RESULTS Compared with controls,the volume of right hippocampus-amygdala transition area(t=-3.053,P=0.003)in patients with T2DM was significantly reduced,which was negatively correlated with the required time of the Trail Making Test(TMT)-A(r=-0.331,P=0.028)and TMT-B(r=-0.402,P=0.007)and positively correlated with the scores of Symbol Digit Modalities Test(r=0.381,P=0.011),Auditory Verbal Learning Test(AVLT)-N7(r=0.309,P=0.041),and Digital Span Test(r=0.300,P=0.048).The volume of the right molecular layer(t=-2.998,P=0.004)was also significantly reduced,which was positively associated with the scores of AVLT-N7(r=0.311,P=0.045).In addition,the left hippocampal fissure volume(t=3.617,P=0.002)was significantly increased in patients with T2DM.CONCLUSION Declines in cognitive performance,especially memory and executive function,are linked to changes in the volumes of the right hippocampus-amygdala transition area and right molecular layer in patients with T2DM.展开更多
AIM： To determine the parameters most informative in predicting the anatomical results of surgical treatment of idiopathic full-thickness macular hole （IMH）. METHODS： One hundred and sixty-two consecutive patien...AIM： To determine the parameters most informative in predicting the anatomical results of surgical treatment of idiopathic full-thickness macular hole （IMH）. METHODS： One hundred and sixty-two consecutive patients （170 eyes） after primary operation for IMH were enrolled. Outcomes were classified as anatomical success when both IMH closure and restoration of the outer retinal structure were achieved. ＂Prospective＂ group included 108 patients （115 eyes） followed with optical coherence tomography （OCT） and microperimetry for ly after surgery. Potential prognostic criteria, except microperimetry data, were tested in ＂retrospective＂ group （54 patients, 55 eyes）. Prognostic value of each parameter was determined using receiver operating characteristic （ROC） analysis. Combined predictive power of the best prognostic parameters was tested with the use of linear discriminant analysis. RESULTS： IMH closure was achieved in 106 eyes （92%） in the prospective group and 49 eyes （89%） in the retrospective group. Despite anatomical closure, the outer retinal structure was not restored in two eyes in the first group and in one eye in the second group. Preoperative central subfield retinal thickness demonstrated the best discriminatory capability between eyes with anatomical success and failure： area under the ROC-curve （AUC） 0.938 （95% Ch 0.881-0.995）, sensitivity 64% at fixed specificity 95% （cut-off value 300um） in the prospective group; sensitivity 57% and specificity 90% in the retrospective group. Other continuous parameters except tractional hole index （AUC： 0.796, 95% Ch 0.591- 1.000） had significantly lower AUCs （P〈0.05）. The best combination of the parameters, established by discriminant analysis in the prospective group, could not confirm its predictive value in the retrospective group. CONCLUSION： Preoperative central subfield retinal thickness is a strong and probably the best predictor of anatomical results of IMH surgical treatment.展开更多
AIM：To investigate the predictive factors for shortterm effects of intravitreal bevacizumab injections on central subfield foveal thickness（CSFT）in patients with macular edema（ME）secondary to central retinal vein...AIM：To investigate the predictive factors for shortterm effects of intravitreal bevacizumab injections on central subfield foveal thickness（CSFT）in patients with macular edema（ME）secondary to central retinal vein occlusion（CRVO）.·METHODS：This was a retrospective study in 60 eyes treated with intravitreal bevacizumab injections for ME due to CRVO.Follow-up was three months.The Early Treatment Diabetic Retinopathy Study（ETDRS）score and CSFT measured by spectral-domain optical coherence tomography（SD-OCT）were used to observe the changes in best-corrected visual acuity（BCVA）.Baseline BCVA,CSFT,age,CRVO duration and the presence of cystoid macular edema（CME）or subretinal fluid（SRF）were analyzed as potential predictive factors of the effects of intravitreal bevacizumab injections.·R ESULTS：BCVA improved from 0.9 log MAR at baseline to 0.6 log MAR at 3mo,which was associated with a significant reduction in CSFT from 721μm to 392μm 3mo after injection.About 50%of CME cases and more than90%of SRF cases responded to treatment with a complete resolution at 3mo.Age（=0.036）and low baseline CSFT（=0.037）were associated with a good 3-month prognosis.Patients〉60 years old achieved better CME resolution（=0.031）and lower CSFT at 3mo（305μm 474μm,=0.003）.·CONCLUSION：Intravitreal bevacizumab significantly improved visual acuity and CSFT in patients with CRVO after 3mo.Older age and lower baseline CSFT were good predictors of short-term CSFT outcomes.The retinal thickness response to bevacizumab might depend on the resolution of CME rather than SRF.展开更多
AIM：To report foveal thickness reduction in eyes with resolution of macular edema and recovery of a foveal depression after one-year of anti-vascular endothelial growth factor（anti-VEGF） therapy for center-involvin...AIM：To report foveal thickness reduction in eyes with resolution of macular edema and recovery of a foveal depression after one-year of anti-vascular endothelial growth factor（anti-VEGF） therapy for center-involving diabetic macular edema（DME）.METHODS：Foveal thickness was assessed with optical coherence tomography to determine the central subfield foveal thickness（CSFT） and macular volume in 42 eyes with DME（CSFT〉275 μm）. Evaluations also included measurement of best-corrected visual acuity（BCVA）, and were performed at baseline, and upon foveal depression recovery achieved after 12 monthly intravitreal injections of either 1.5 mg/0.06 mL bevacizumab（n=21） or 0.5 mg/0.05 mL ranibizumab（n=21）. Data was compared to 42 eyes of normally sighted, non-diabetic, healthy individuals with similar age, gender and race distributions.RESULTS：Mean baseline BCVA was 0.59±0.04 and 0.32± 0.03 log MAR（P〈0.001） after treatment and resolution of DME, with all, but 3 eyes, showing BCVA improvement. Mean CSFT before treatment was 422.0±20.0 μm, and after treatment, decreased to 241.6±4.6 μm（P〈0.001）, which is significantly thinner than CSFT found in control subjects（272.0±3.4 μm; P〈0.001）. Moreover, in 33/42 DM eyes（79%）, CSTF was thinner than the matched control eye. Macular volume showed comparable results, but with lower differences between groups（control：8.5±0.4 mm^3; DME：8.2±1.0 mm^3; P=0.0267）.CONCLUSION：DME eyes show significantly lower foveal thickness than matched controls after DME resolution achieved with one-year anti-VEGF therapy. Further investigation into the reasonsfor this presumable retinal atrophy using fluorescein angiography and functional parameters as well as establishing possible predictors is warranted. This finding should be considered during the treatment of DME.展开更多
In this work, we discuss the possibility to classify relativity in accordance with the classification of second order partial differential equations that have been applied into the formulation of physical laws in phys...In this work, we discuss the possibility to classify relativity in accordance with the classification of second order partial differential equations that have been applied into the formulation of physical laws in physics. In mathematics, since second order partial differential equations can be classified into hyperbolic, elliptic or parabolic type, therefore we show that it is also possible to classify relativity accordingly into hyperbolic, elliptic or parabolic type by establishing coordinate transformations that preserve the forms of these second order partial differential equations. The coordinate transformation that preserves the form of the hyperbolic equation is the Lorentz transformation and the associated space is the hyperbolic, or pseudo-Euclidean, relativistic spacetime. Typical equations in physics that comply with hyperbolic relativity are Maxwell and Dirac equations. The coordinate transformation that preserves the form of the elliptic equation is the modified Lorentz transformation that we have formulated in our work on Euclidean relativity and the associated space is the elliptic, or Euclidean, relativistic spacetime. As we will show in this work, equations that comply with elliptic relativity are the equations that describe the subfields of Maxwell and Dirac field. And the coordinate transformation that preserves the form of the parabolic equation is the Euclidean transformation consisting of the translation and rotation in the spatial space and the associated space is the parabolic relativistic spacetime, which is a Euclidean space with a universal time. Typical equations in physics that comply with parabolic relativity are the diffusion equation, the Schrödinger equation and in particular the diffusion equations that are derived from the four-current defined in terms of the differentiable structures of the spacetime manifold, and the Ricci flow.展开更多
Objective In order to red uc e the flicker in color plasma display panel (PDP), the highly peaked light outpu t at certain positions and the strong luminance difference among subfields, whic h cause flicker in PDP, i...Objective In order to red uc e the flicker in color plasma display panel (PDP), the highly peaked light outpu t at certain positions and the strong luminance difference among subfields, whic h cause flicker in PDP, is analyzed; and a flicker index method is developed to evaluate flicker in PDP based on the flicker theory and the electroencephalograp h response model for luminance. Methods This method considers f licker as a main result of the fundamental frequency component of luminance supe rimposed on the DC-component, the flicker index is defined as ratio between the positive square roots of the sum of the two square components. Results & Conclusion The calculation results of flicker index indicate that flicker can be obviously weakened via increase of subfield number, symmetrical s election and homogeneous interlaced distribution of major weight subfields . Fli cker can be further reduced by choosing the suitable subfield order according to gray scale region of image.展开更多
The authors exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers.They also prove the 3-divisibility of the plus part of the class number of another family consisting ...The authors exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers.They also prove the 3-divisibility of the plus part of the class number of another family consisting of infinitely many cyclotomic fields.At the end,they provide some numerical examples supporting our results.展开更多
Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurrin...Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .展开更多
Assume that K is a number field, L<sub>i</sub>=K(D<sub>i</sub><sup>1/2</sup>)（i=1, 2）, where D<sub>1</sub>, D<sub>2</sub>∈K\K<sup>2</sup> and ...Assume that K is a number field, L<sub>i</sub>=K(D<sub>i</sub><sup>1/2</sup>)（i=1, 2）, where D<sub>1</sub>, D<sub>2</sub>∈K\K<sup>2</sup> and L<sub>1</sub>≠L<sub>2</sub>. Let L=L<sub>1</sub>L<sub>2</sub>. It is well-known that L/K has three intermediate subfields of degree 2, which are L<sub>1</sub>, L<sub>2</sub> and L3=K(D<sub>1</sub>D<sub>2</sub><sup>1/2</sup>).展开更多
Let F_(p)^(m) be a finite field with p^(m) elements,where p is an odd prime and m is a positive integer.Recently,[17]and[35]determined the weight distributions of subfield codes with the form C f={((T r(a f(x)+b x)+c)...Let F_(p)^(m) be a finite field with p^(m) elements,where p is an odd prime and m is a positive integer.Recently,[17]and[35]determined the weight distributions of subfield codes with the form C f={((T r(a f(x)+b x)+c)_(x∈F_(p)^(m)),T r(a)):a,b∈F_(p)^(m),c∈F_(p)}for f(x)=x^(2) and f(x)=x p k+1,respectively,where Tr(⋅)is the trace function from F_(p)^(m) to F_(p),and k is a nonnegative integer.In this paper,we further investigate the subfield code C f for f(x)being a known perfect nonlinear function over F_(p)^(m) and generalize some results in[17,35].The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields.In addition,the parameters of the duals of these codes are also determined.Several examples show that some of our codes and their duals have the best known parameters according to the code tables in[16].The duals of some proposed codes are optimal according to the Sphere Packing bound if p≥5.展开更多
基金Supported by the National Natural Science Foundation of China(Nos.60971133,61271112)
文摘The volume of hippocampal subfields is closely related with early diagnosis of Alzheimer's disease.Due to the anatomical complexity of hippocampal subfields,automatic segmentation merely on the content of MR images is extremely difficult.We presented a method which combines multi-atlas image segmentation with extreme learning machine based bias detection and correction technique to achieve a fully automatic segmentation of hippocampal subfields.Symmetric diffeomorphic registration driven by symmetric mutual information energy was implemented in atlas registration,which allows multi-modal image registration and accelerates execution time.An exponential function based label fusion strategy was proposed for the normalized similarity measure case in segmentation combination,which yields better combination accuracy.The test results show that this method is effective,especially for the larger subfields with an overlap of more than 80%,which is competitive with the current methods and is of potential clinical significance.
基金Supported by the Bethune Charitable Foundation,No.Z04JKM2022E035the Liyang City's 2023 Annual research and development Plan Follows Nanjing Project,No.LC2024001.
文摘BACKGROUND Increasing evidence has shown that hippocampal damage serves as a marker of early cognitive decline in patients with type 2 diabetes mellitus(T2DM);however,the association between hippocampal subregion volume changes and cognitive decline in different dimensions remains unclear.AIM To investigate changes in hippocampal subregion volumes in patients with T2DM and their relationship with cognitive function impairment.METHODS Sixty patients with T2DM and 32 healthy controls were recruited.All participants underwent a 3.0 T magnetic resonance scan and a series of clinical assessments.Hippocampal subfield volumes were determined using FreeSurfer 7.4.1.A two-sample t-test was used to evaluate group differences.Partial correlation analysis was performed to assess the relationship between hippocampal subregion volumes and cognitive function.aP<0.05 was considered statistically significant.RESULTS Compared with controls,the volume of right hippocampus-amygdala transition area(t=-3.053,P=0.003)in patients with T2DM was significantly reduced,which was negatively correlated with the required time of the Trail Making Test(TMT)-A(r=-0.331,P=0.028)and TMT-B(r=-0.402,P=0.007)and positively correlated with the scores of Symbol Digit Modalities Test(r=0.381,P=0.011),Auditory Verbal Learning Test(AVLT)-N7(r=0.309,P=0.041),and Digital Span Test(r=0.300,P=0.048).The volume of the right molecular layer(t=-2.998,P=0.004)was also significantly reduced,which was positively associated with the scores of AVLT-N7(r=0.311,P=0.045).In addition,the left hippocampal fissure volume(t=3.617,P=0.002)was significantly increased in patients with T2DM.CONCLUSION Declines in cognitive performance,especially memory and executive function,are linked to changes in the volumes of the right hippocampus-amygdala transition area and right molecular layer in patients with T2DM.
文摘AIM： To determine the parameters most informative in predicting the anatomical results of surgical treatment of idiopathic full-thickness macular hole （IMH）. METHODS： One hundred and sixty-two consecutive patients （170 eyes） after primary operation for IMH were enrolled. Outcomes were classified as anatomical success when both IMH closure and restoration of the outer retinal structure were achieved. ＂Prospective＂ group included 108 patients （115 eyes） followed with optical coherence tomography （OCT） and microperimetry for ly after surgery. Potential prognostic criteria, except microperimetry data, were tested in ＂retrospective＂ group （54 patients, 55 eyes）. Prognostic value of each parameter was determined using receiver operating characteristic （ROC） analysis. Combined predictive power of the best prognostic parameters was tested with the use of linear discriminant analysis. RESULTS： IMH closure was achieved in 106 eyes （92%） in the prospective group and 49 eyes （89%） in the retrospective group. Despite anatomical closure, the outer retinal structure was not restored in two eyes in the first group and in one eye in the second group. Preoperative central subfield retinal thickness demonstrated the best discriminatory capability between eyes with anatomical success and failure： area under the ROC-curve （AUC） 0.938 （95% Ch 0.881-0.995）, sensitivity 64% at fixed specificity 95% （cut-off value 300um） in the prospective group; sensitivity 57% and specificity 90% in the retrospective group. Other continuous parameters except tractional hole index （AUC： 0.796, 95% Ch 0.591- 1.000） had significantly lower AUCs （P〈0.05）. The best combination of the parameters, established by discriminant analysis in the prospective group, could not confirm its predictive value in the retrospective group. CONCLUSION： Preoperative central subfield retinal thickness is a strong and probably the best predictor of anatomical results of IMH surgical treatment.
文摘AIM：To investigate the predictive factors for shortterm effects of intravitreal bevacizumab injections on central subfield foveal thickness（CSFT）in patients with macular edema（ME）secondary to central retinal vein occlusion（CRVO）.·METHODS：This was a retrospective study in 60 eyes treated with intravitreal bevacizumab injections for ME due to CRVO.Follow-up was three months.The Early Treatment Diabetic Retinopathy Study（ETDRS）score and CSFT measured by spectral-domain optical coherence tomography（SD-OCT）were used to observe the changes in best-corrected visual acuity（BCVA）.Baseline BCVA,CSFT,age,CRVO duration and the presence of cystoid macular edema（CME）or subretinal fluid（SRF）were analyzed as potential predictive factors of the effects of intravitreal bevacizumab injections.·R ESULTS：BCVA improved from 0.9 log MAR at baseline to 0.6 log MAR at 3mo,which was associated with a significant reduction in CSFT from 721μm to 392μm 3mo after injection.About 50%of CME cases and more than90%of SRF cases responded to treatment with a complete resolution at 3mo.Age（=0.036）and low baseline CSFT（=0.037）were associated with a good 3-month prognosis.Patients〉60 years old achieved better CME resolution（=0.031）and lower CSFT at 3mo（305μm 474μm,=0.003）.·CONCLUSION：Intravitreal bevacizumab significantly improved visual acuity and CSFT in patients with CRVO after 3mo.Older age and lower baseline CSFT were good predictors of short-term CSFT outcomes.The retinal thickness response to bevacizumab might depend on the resolution of CME rather than SRF.
基金Supported by Fundacao de Amparoà Pesquisa do Estado de Sao Paulo(FAPESP)and FAEPA(Fundacao Apoioao Ensino Pesquisa e Assistência,HCFMRP-USP),(No.2010/013368)the initial trial was registered at clinical trials.gov(No.NCT01487629)
文摘AIM：To report foveal thickness reduction in eyes with resolution of macular edema and recovery of a foveal depression after one-year of anti-vascular endothelial growth factor（anti-VEGF） therapy for center-involving diabetic macular edema（DME）.METHODS：Foveal thickness was assessed with optical coherence tomography to determine the central subfield foveal thickness（CSFT） and macular volume in 42 eyes with DME（CSFT〉275 μm）. Evaluations also included measurement of best-corrected visual acuity（BCVA）, and were performed at baseline, and upon foveal depression recovery achieved after 12 monthly intravitreal injections of either 1.5 mg/0.06 mL bevacizumab（n=21） or 0.5 mg/0.05 mL ranibizumab（n=21）. Data was compared to 42 eyes of normally sighted, non-diabetic, healthy individuals with similar age, gender and race distributions.RESULTS：Mean baseline BCVA was 0.59±0.04 and 0.32± 0.03 log MAR（P〈0.001） after treatment and resolution of DME, with all, but 3 eyes, showing BCVA improvement. Mean CSFT before treatment was 422.0±20.0 μm, and after treatment, decreased to 241.6±4.6 μm（P〈0.001）, which is significantly thinner than CSFT found in control subjects（272.0±3.4 μm; P〈0.001）. Moreover, in 33/42 DM eyes（79%）, CSTF was thinner than the matched control eye. Macular volume showed comparable results, but with lower differences between groups（control：8.5±0.4 mm^3; DME：8.2±1.0 mm^3; P=0.0267）.CONCLUSION：DME eyes show significantly lower foveal thickness than matched controls after DME resolution achieved with one-year anti-VEGF therapy. Further investigation into the reasonsfor this presumable retinal atrophy using fluorescein angiography and functional parameters as well as establishing possible predictors is warranted. This finding should be considered during the treatment of DME.
文摘In this work, we discuss the possibility to classify relativity in accordance with the classification of second order partial differential equations that have been applied into the formulation of physical laws in physics. In mathematics, since second order partial differential equations can be classified into hyperbolic, elliptic or parabolic type, therefore we show that it is also possible to classify relativity accordingly into hyperbolic, elliptic or parabolic type by establishing coordinate transformations that preserve the forms of these second order partial differential equations. The coordinate transformation that preserves the form of the hyperbolic equation is the Lorentz transformation and the associated space is the hyperbolic, or pseudo-Euclidean, relativistic spacetime. Typical equations in physics that comply with hyperbolic relativity are Maxwell and Dirac equations. The coordinate transformation that preserves the form of the elliptic equation is the modified Lorentz transformation that we have formulated in our work on Euclidean relativity and the associated space is the elliptic, or Euclidean, relativistic spacetime. As we will show in this work, equations that comply with elliptic relativity are the equations that describe the subfields of Maxwell and Dirac field. And the coordinate transformation that preserves the form of the parabolic equation is the Euclidean transformation consisting of the translation and rotation in the spatial space and the associated space is the parabolic relativistic spacetime, which is a Euclidean space with a universal time. Typical equations in physics that comply with parabolic relativity are the diffusion equation, the Schrödinger equation and in particular the diffusion equations that are derived from the four-current defined in terms of the differentiable structures of the spacetime manifold, and the Ricci flow.
基金Thisprojectwassupportedbythe0205ProjectforScienceandTechnology of china edncation ministry.
文摘Objective In order to red uc e the flicker in color plasma display panel (PDP), the highly peaked light outpu t at certain positions and the strong luminance difference among subfields, whic h cause flicker in PDP, is analyzed; and a flicker index method is developed to evaluate flicker in PDP based on the flicker theory and the electroencephalograp h response model for luminance. Methods This method considers f licker as a main result of the fundamental frequency component of luminance supe rimposed on the DC-component, the flicker index is defined as ratio between the positive square roots of the sum of the two square components. Results & Conclusion The calculation results of flicker index indicate that flicker can be obviously weakened via increase of subfield number, symmetrical s election and homogeneous interlaced distribution of major weight subfields . Fli cker can be further reduced by choosing the suitable subfield order according to gray scale region of image.
基金supported by the SERB MATRICS Grant(No.MTR/2021/00762)the ANRF(SERB)Govt.of India Grant(No.CRG/2023/007323).
文摘The authors exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers.They also prove the 3-divisibility of the plus part of the class number of another family consisting of infinitely many cyclotomic fields.At the end,they provide some numerical examples supporting our results.
基金supported by National Key Basic Research Program of China(973 Program)(Grant No.2013CB834204)National Natural Science Foundation of China(Grant Nos.61171082 and 10990011)
文摘Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .
基金Project supported by the National Natural Science Foundation of China.
文摘Assume that K is a number field, L<sub>i</sub>=K(D<sub>i</sub><sup>1/2</sup>)（i=1, 2）, where D<sub>1</sub>, D<sub>2</sub>∈K\K<sup>2</sup> and L<sub>1</sub>≠L<sub>2</sub>. Let L=L<sub>1</sub>L<sub>2</sub>. It is well-known that L/K has three intermediate subfields of degree 2, which are L<sub>1</sub>, L<sub>2</sub> and L3=K(D<sub>1</sub>D<sub>2</sub><sup>1/2</sup>).
基金This work was supported in part by the National Natural Science Foundation of China(NSFC)under Grants 11971156 and 12001175.
文摘Let F_(p)^(m) be a finite field with p^(m) elements,where p is an odd prime and m is a positive integer.Recently,[17]and[35]determined the weight distributions of subfield codes with the form C f={((T r(a f(x)+b x)+c)_(x∈F_(p)^(m)),T r(a)):a,b∈F_(p)^(m),c∈F_(p)}for f(x)=x^(2) and f(x)=x p k+1,respectively,where Tr(⋅)is the trace function from F_(p)^(m) to F_(p),and k is a nonnegative integer.In this paper,we further investigate the subfield code C f for f(x)being a known perfect nonlinear function over F_(p)^(m) and generalize some results in[17,35].The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields.In addition,the parameters of the duals of these codes are also determined.Several examples show that some of our codes and their duals have the best known parameters according to the code tables in[16].The duals of some proposed codes are optimal according to the Sphere Packing bound if p≥5.
