1. (Currently Amended) An encoder which encodes a current block to be encoded in an image,
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`AMENDMENTS TO THE CLAIMS
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`the encoder comprising:
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`circuitry; and
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`memory,
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`wherein using the memory,the circuitry:
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`performs a primary transform from residuals of the current block to primary coefficients;
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`determines whether a secondary transform is to be applied to the current block;
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`(i) calculates quantized primary coefficients by performing a first quantization on the
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`primary coefficients when the secondary transform is not to be applied, (11) performs a secondary
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`transform from the primary coefficients to secondary coefficients, and calculates quantized
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`secondary coefficients by performing a secondseeendary quantization on the secondary
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`coefficients when the secondary transform is to be applied, the secondsecendary quantization
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`being different from the first quantization; and
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`generates an encodedbitstream by encoding either the quantized primary coefficients or
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`the quantized secondary coefficients.
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`2. (Original) The encoder according to claim 1,
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`wherein the first quantization is a weighted quantization using a first quantization matrix,
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`and
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`the second quantization is a weighted quantization using a second quantization matrix
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`different from the first quantization matrix.
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`3. (Original) The encoder according to claim 2,
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`wherein the circuitry writes the first quantization matrix and the second quantization
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`matrix into the encoded bitstream.
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`4. (Original) The encoder according to claim 3,
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`wherein the primary coefficients include one or morefirst primary coefficients and one or
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`more second primary coefficients,
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`

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`the secondary transform is applied to the one or morefirst primary coefficients and is not
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`applied to the one or more second primary coefficients,
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`the second quantization matrix has one or more first component values corresponding to
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`the one or morefirst primary coefficients and one or more second component values
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`corresponding to the one or more second primary coefficients,
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`each of the one or more second component values of the second quantization matrix
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`matches a corresponding one of componentvaluesofthe first quantization matrix, and
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`whenthe circuitry writes the second quantization matrix, the circuitry writes, into the
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`encodedbitstream, only the one or more first component values among the one or morefirst
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`component values and the one or more second componentvalues.
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`5. (Original) The encoder accordingto claim 3,
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`wherein in the secondary transform, a plurality of bases which has been determined is
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`selectively used,
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`the encoded bitstream includes a plurality of second quantization matrices corresponding
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`to the plurality of bases, and
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`in the second quantization, a second quantization matrix corresponding to a basis used for
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`the second transform is selected from the plurality of second quantization matrices.
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`6. (Original) The encoder according to claim 2,
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`wherein the first quantization matrix and the second quantization matrix are defined in
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`advance in a standard.
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`7. (Original) The encoder according to claim 2,
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`wherein the circuitry derives the second quantization matrix from the first quantization
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`matrix.
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`8. (Original) The encoder accordingto claim 7,
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`wherein the primary coefficients include one or morefirst primary coefficients and one or
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`more second primary coefficients,
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`

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`the secondary transform is applied to the one or morefirst primary coefficients and is not
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`applied to the one or more second primary coefficients,
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`the second quantization matrix has one or more first component values corresponding to
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`the one or morefirst primary coefficients and one or more second component values
`
`corresponding to the one or more second primary coefficients,
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`each of the one or more second component values of the second quantization matrix
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`matches a corresponding one of componentvaluesofthe first quantization matrix, and
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`whenthecircuitry derives the second quantization matrix, the circuitry derives the one or
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`more first componentvalues of the second quantization matrix from the first quantization matrix.
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`9. (Original) The encoder according to claim 7,
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`wherein the second quantization matrix is derived by applying the secondary transform to
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`the first quantization matrix.
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`10. (Original) The encoder accordingto claim 7,
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`wherein the circuitry:
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`derives a third quantization matrix from the first quantization matrix, the third
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`quantization matrix having component values which are greater when corresponding component
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`values ofthe first quantization matrix are smaller;
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`derives a fourth quantization matrix by applying the secondary transform to the third
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`quantization matrix; and
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`derives, as the second quantization matrix, a fifth quantization matrix from the fourth
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`quantization matrix, the fifth quantization matrix having component values which are greater
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`when corresponding componentvalues of the fourth quantization matrix are smaller.
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`11. (Original) The encoder according to claim 10,
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`wherein the componentvalues of the third quantization matrix are reciprocals of the
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`corresponding componentvaluesofthe first quantization matrix, and
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`the componentvalues ofthe fifth quantization matrix are reciprocals of the corresponding
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`component values of the fourth quantization matrix.
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`

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`12. (Original) The encoderaccordingto claim 1,
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`wherein the first quantization is a weighted quantization in which a quantization matrix is
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`used, and
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`the second quantization is a non-weighted quantization in which no quantization matrix is
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`used.
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`13. (Original) The encoder accordingto claim 1,
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`wherein the first quantization is a weighted quantization using a first quantization matrix,
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`in the secondary transform,(i) weighted primary coefficients are calculated by
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`multiplying each of the primary coefficients by a corresponding componentvalue of a weighting
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`matrix, and (ii) the weighted primary coefficients are transformed into secondary coefficients,
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`and
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`in the second quantization, each of the secondary coefficients is divided by a quantization
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`step commonbetween the secondary coefficients.
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`14. (Original) The encoder according to claim 13,
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`wherein the circuitry derives the weighting matrix from the first quantization matrix.
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`15. (Original) The encoderaccording to claim 13,
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`wherein the circuitry derives the common quantization step from a quantization
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`parameter for the current block.
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`16. (Currently Amended) An encoding method for encoding a current block to be encoded in an
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`image, the encoding method comprising:
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`performing a primary transform from residuals of the current block to primary
`
`coefficients;
`
`determining whether a secondary transform is to be applied to the current block;
`
`(i) calculating quantized primary coefficients by performing a first quantization on the
`
`primary coefficients when the secondary transform is not to be applied, (11) performing a
`
`secondary transform from the primary coefficients to secondary coefficients, and calculating
`
`quantized secondary coefficients by performing a secondsecerndary quantization on the
`
`

`

`secondary coefficients when the secondary transform is to be applied, the secondsecerdary
`
`quantization being different from the first quantization; and
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`generating an encodedbitstream by encoding either the quantized primary coefficients or
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`the quantized secondary coefficients.
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`17. (Withdrawn) An encoder which encodesa current block to be encodedin an image, the
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`encoder comprising:
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`circuitry; and
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`memory,
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`wherein using the memory,the circuitry:
`
`performs a primary transform on residuals of the current block to primary coefficients;
`
`determines whether a secondary transform is to be applied to the current block;
`
`(i) calculates first quantized primary coefficients by performing a first quantization on the
`
`primary coefficients when the secondary transform is not to be applied, (11) calculates second
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`quantized primary coefficients by performing a second quantization onthefirst coefficients, and
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`performs a secondary transform from second quantized primary coefficients to quantized
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`secondary coefficients when the secondary transform is to be applied; and
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`generates an encodedbitstream by encoding either the quantized primary coefficients or
`
`the quantized secondary coefficients.
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`18. (Withdrawn) An encoding method for encoding a current block to be encodedin an image,
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`the encoding method comprising:
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`performing a primary transform on residuals of the current block to primary coefficients;
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`determining whether a secondary transform is to be applied to the current block;
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`(i) calculating first quantized primary coefficients by performinga first quantization on
`
`the primary coefficients when the secondary transform is not to be applied, (ii) calculating
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`second quantized primary coefficients by performing a second quantization on thefirst
`
`coefficients, and performing a secondary transform from second quantized primary coefficients
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`to quantized secondary coefficients when the secondary transform is to be applied; and
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`generating an encodedbitstream by encoding either the quantized primary coefficients or
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`the quantized secondary coefficients.
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`

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`19 — 36. (Cancelled)
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`37. (New) The encoder according to claim 1, wherein the first quantization uses a quantization
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`matrix, and the second quantization uses no quantization matrix.
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`38. (New) The encoderaccording to claim 1, wherein the second quantization divides each
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`secondary coefficient of the current block by a commoninteger.
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`39. (New) The encoding method according to claim 16, wherein the first quantization uses a
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`quantization matrix, and the second quantization uses no quantization matrix.
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`40. (New) The encoding method according to claim 16, wherein the second quantization divides
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`each secondary coefficient of the current block by a commoninteger.
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`41. (New) An encoder comprising:
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`memory; and
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`circuitry coupled to the memory and configuredto:
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`perform a primary transform and a secondary transform to convert residuals in a
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`current block to coefficients;
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`quantize the coefficients with no quantization matrix to generate quantized
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`coefficients; and
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`encode the quantized coefficients to generate an encodedbitstream.
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